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case inl n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L hS : Set.Nonempty S ⊢ IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm ·
exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm ·
Mathlib.NumberTheory.Cyclotomic.Basic.251_0.xReI1DeVvechFQU
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B
Mathlib_NumberTheory_Cyclotomic_Basic
case inr n : ℕ+ T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L ⊢ IsCyclotomicExtension ∅ A B ↔ IsCyclotomicExtension (∅ ∪ {1}) A B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
Mathlib.NumberTheory.Cyclotomic.Basic.251_0.xReI1DeVvechFQU
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B
Mathlib_NumberTheory_Cyclotomic_Basic
case inr n : ℕ+ T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L ⊢ IsCyclotomicExtension ∅ A B ↔ IsCyclotomicExtension {1} A B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union]
Mathlib.NumberTheory.Cyclotomic.Basic.251_0.xReI1DeVvechFQU
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B
Mathlib_NumberTheory_Cyclotomic_Basic
case inr.refine'_1 n : ℕ+ T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L H : IsCyclotomicExtension ∅ A B ⊢ IsCyclotomicExtension {1} A B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ ·
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ ·
Mathlib.NumberTheory.Cyclotomic.Basic.251_0.xReI1DeVvechFQU
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L H : IsCyclotomicExtension ∅ A B s : ℕ+ hs : s ∈ {1} ⊢ IsPrimitiveRoot 1 ↑s
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by
simp [mem_singleton_iff.1 hs]
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by
Mathlib.NumberTheory.Cyclotomic.Basic.251_0.xReI1DeVvechFQU
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B
Mathlib_NumberTheory_Cyclotomic_Basic
case inr.refine'_1 n : ℕ+ T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L H : IsCyclotomicExtension ∅ A B ⊢ adjoin A {b | ∃ n ∈ {1}, b ^ ↑n = 1} = ⊤
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
Mathlib.NumberTheory.Cyclotomic.Basic.251_0.xReI1DeVvechFQU
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B
Mathlib_NumberTheory_Cyclotomic_Basic
case inr.refine'_2 n : ℕ+ T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L H : IsCyclotomicExtension {1} A B ⊢ IsCyclotomicExtension ∅ A B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] ·
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] ·
Mathlib.NumberTheory.Cyclotomic.Basic.251_0.xReI1DeVvechFQU
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B
Mathlib_NumberTheory_Cyclotomic_Basic
case inr.refine'_2 n : ℕ+ T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L H : IsCyclotomicExtension {1} A B ⊢ adjoin A {b | ∃ n ∈ ∅, b ^ ↑n = 1} = ⊤
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
Mathlib.NumberTheory.Cyclotomic.Basic.251_0.xReI1DeVvechFQU
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L h : ⊥ = ⊤ ⊢ IsCyclotomicExtension {1} A B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by
Mathlib.NumberTheory.Cyclotomic.Basic.266_0.xReI1DeVvechFQU
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B
Mathlib_NumberTheory_Cyclotomic_Basic
case h.e'_1 n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L h : ⊥ = ⊤ ⊢ {1} = ∅ ∪ {1}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
Mathlib.NumberTheory.Cyclotomic.Basic.266_0.xReI1DeVvechFQU
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁷ : CommRing A inst✝⁶ : CommRing B inst✝⁵ : Algebra A B inst✝⁴ : Field K inst✝³ : Field L inst✝² : Algebra K L C : Type u_1 inst✝¹ : CommRing C inst✝ : Algebra A C h : IsCyclotomicExtension S A B f : B ≃ₐ[A] C ⊢ IsCyclotomicExtension S A C
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
Mathlib.NumberTheory.Cyclotomic.Basic.281_0.xReI1DeVvechFQU
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁷ : CommRing A inst✝⁶ : CommRing B inst✝⁵ : Algebra A B inst✝⁴ : Field K inst✝³ : Field L inst✝² : Algebra K L C : Type u_1 inst✝¹ : CommRing C inst✝ : Algebra A C h : IsCyclotomicExtension S A B f : B ≃ₐ[A] C this : Algebra B C := RingHom.toAlgebra ↑↑f ⊢ IsCyclotomicExtension S A C
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
Mathlib.NumberTheory.Cyclotomic.Basic.281_0.xReI1DeVvechFQU
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁷ : CommRing A inst✝⁶ : CommRing B inst✝⁵ : Algebra A B inst✝⁴ : Field K inst✝³ : Field L inst✝² : Algebra K L C : Type u_1 inst✝¹ : CommRing C inst✝ : Algebra A C h : IsCyclotomicExtension S A B f : B ≃ₐ[A] C this✝ : Algebra B C := RingHom.toAlgebra ↑↑f this : IsCyclotomicExtension {1} B C ⊢ IsCyclotomicExtension S A C
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
Mathlib.NumberTheory.Cyclotomic.Basic.281_0.xReI1DeVvechFQU
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁷ : CommRing A inst✝⁶ : CommRing B inst✝⁵ : Algebra A B inst✝⁴ : Field K inst✝³ : Field L inst✝² : Algebra K L C : Type u_1 inst✝¹ : CommRing C inst✝ : Algebra A C h : IsCyclotomicExtension S A B f : B ≃ₐ[A] C this✝¹ : Algebra B C := RingHom.toAlgebra ↑↑f this✝ : IsCyclotomicExtension {1} B C this : IsScalarTower A B C ⊢ IsCyclotomicExtension S A C
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
Mathlib.NumberTheory.Cyclotomic.Basic.281_0.xReI1DeVvechFQU
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L h : IsCyclotomicExtension {n} A B inst✝ : IsDomain B ⊢ NeZero ↑↑n
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
Mathlib.NumberTheory.Cyclotomic.Basic.292_0.xReI1DeVvechFQU
protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B)
Mathlib_NumberTheory_Cyclotomic_Basic
case intro.intro n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L h : IsCyclotomicExtension {n} A B inst✝ : IsDomain B r : B hr : IsPrimitiveRoot r ↑n ⊢ NeZero ↑↑n
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
Mathlib.NumberTheory.Cyclotomic.Basic.292_0.xReI1DeVvechFQU
protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B)
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁷ : CommRing A inst✝⁶ : CommRing B inst✝⁵ : Algebra A B inst✝⁴ : Field K inst✝³ : Field L inst✝² : Algebra K L inst✝¹ : IsCyclotomicExtension {n} A B inst✝ : IsDomain B ⊢ NeZero ↑↑n
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
Mathlib.NumberTheory.Cyclotomic.Basic.298_0.xReI1DeVvechFQU
protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A)
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁷ : CommRing A inst✝⁶ : CommRing B inst✝⁵ : Algebra A B inst✝⁴ : Field K inst✝³ : Field L inst✝² : Algebra K L inst✝¹ : IsCyclotomicExtension {n} A B inst✝ : IsDomain B this : NeZero ↑↑n ⊢ NeZero ↑↑n
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B
Mathlib.NumberTheory.Cyclotomic.Basic.298_0.xReI1DeVvechFQU
protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A)
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B h : IsCyclotomicExtension {n} A B ⊢ Module.Finite A B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by
classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by
Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B h : IsCyclotomicExtension {n} A B ⊢ Module.Finite A B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical
Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B h : IsCyclotomicExtension {n} A B ⊢ Submodule.FG (Subalgebra.toSubmodule (adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1}))
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_1 n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B h : IsCyclotomicExtension {n} A B ⊢ Set.Finite {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ ·
simp only [mem_singleton_iff, exists_eq_left]
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ ·
Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_1 n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B h : IsCyclotomicExtension {n} A B ⊢ Set.Finite {b | b ^ ↑n = 1}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left]
Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B h✝ : IsCyclotomicExtension {n} A B x : B h : x ∈ {b | b ^ ↑n = 1} ⊢ x ∈ ↑(Multiset.toFinset (nthRoots (↑n) 1))
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by
simpa using h
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by
Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B h✝ : IsCyclotomicExtension {n} A B x : B h : x ∈ ↑(Multiset.toFinset (nthRoots (↑n) 1)) ⊢ x ∈ {b | b ^ ↑n = 1}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by
simpa using h
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by
Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_1 n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B h : IsCyclotomicExtension {n} A B this : {b | b ^ ↑n = 1} = ↑(Multiset.toFinset (nthRoots (↑n) 1)) ⊢ Set.Finite {b | b ^ ↑n = 1}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_1 n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B h : IsCyclotomicExtension {n} A B this : {b | b ^ ↑n = 1} = ↑(Multiset.toFinset (nthRoots (↑n) 1)) ⊢ Set.Finite ↑(Multiset.toFinset (nthRoots (↑n) 1))
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this]
Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_2 n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B h : IsCyclotomicExtension {n} A B b : B hb : b ∈ {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} ⊢ IsIntegral A b
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet ·
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet ·
Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_2 n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B h : IsCyclotomicExtension {n} A B b : B hb : b ^ ↑n = 1 ⊢ IsIntegral A b
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B h : IsCyclotomicExtension {n} A B b : B hb : b ^ ↑n = 1 ⊢ eval₂ (algebraMap A B) b (X ^ ↑n - 1) = 0
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by
simp [hb]
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by
Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B h₁ : Finite ↑S h₂ : IsCyclotomicExtension S A B ⊢ Module.Finite A B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by
cases' nonempty_fintype S with h
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by
Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
case intro n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B h₁ : Finite ↑S h₂ : IsCyclotomicExtension S A B h : Fintype ↑S ⊢ Module.Finite A B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h
revert h₂ A B
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h
Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
case intro n : ℕ+ S T : Set ℕ+ K : Type w L : Type z inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L h₁ : Finite ↑S h : Fintype ↑S ⊢ ∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B] [h₂ : IsCyclotomicExtension S A B], Module.Finite A B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B
Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
case intro.refine'_1 n : ℕ+ S T : Set ℕ+ K : Type w L : Type z inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L h₁ : Finite ↑S h : Fintype ↑S A : Type u B : Type v ⊢ ∀ [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B] [h₂ : IsCyclotomicExtension ∅ A B], Module.Finite A B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ ·
intro _ _ _ _ _
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ ·
Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
case intro.refine'_1 n : ℕ+ S T : Set ℕ+ K : Type w L : Type z inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L h₁ : Finite ↑S h : Fintype ↑S A : Type u B : Type v inst✝³ : CommRing A inst✝² : CommRing B inst✝¹ : Algebra A B inst✝ : IsDomain B h₂✝ : IsCyclotomicExtension ∅ A B ⊢ Module.Finite A B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _
Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
case intro.refine'_1 n : ℕ+ S T : Set ℕ+ K : Type w L : Type z inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L h₁ : Finite ↑S h : Fintype ↑S A : Type u B : Type v inst✝³ : CommRing A inst✝² : CommRing B inst✝¹ : Algebra A B inst✝ : IsDomain B h₂✝ : IsCyclotomicExtension ∅ A B ⊢ Submodule.span A ↑{1} = ⊤
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
case intro.refine'_2 n✝ : ℕ+ S✝ T : Set ℕ+ K : Type w L : Type z inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L h₁ : Finite ↑S✝ h : Fintype ↑S✝ n : ℕ+ S : Set ℕ+ x✝¹ : n ∉ S x✝ : Set.Finite S H : ∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B] [h₂ : IsCyclotomicExtension S A B], Module.Finite A B A : Type u B : Type v ⊢ ∀ [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B] [h₂ : IsCyclotomicExtension (insert n S) A B], Module.Finite A B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] ·
intro _ _ _ _ h
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] ·
Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
case intro.refine'_2 n✝ : ℕ+ S✝ T : Set ℕ+ K : Type w L : Type z inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L h₁ : Finite ↑S✝ h✝ : Fintype ↑S✝ n : ℕ+ S : Set ℕ+ x✝¹ : n ∉ S x✝ : Set.Finite S H : ∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B] [h₂ : IsCyclotomicExtension S A B], Module.Finite A B A : Type u B : Type v inst✝³ : CommRing A inst✝² : CommRing B inst✝¹ : Algebra A B inst✝ : IsDomain B h : IsCyclotomicExtension (insert n S) A B ⊢ Module.Finite A B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S)
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h
Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
case intro.refine'_2 n✝ : ℕ+ S✝ T : Set ℕ+ K : Type w L : Type z inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L h₁ : Finite ↑S✝ h✝ : Fintype ↑S✝ n : ℕ+ S : Set ℕ+ x✝¹ : n ∉ S x✝ : Set.Finite S H : ∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B] [h₂ : IsCyclotomicExtension S A B], Module.Finite A B A : Type u B : Type v inst✝³ : CommRing A inst✝² : CommRing B inst✝¹ : Algebra A B inst✝ : IsDomain B h : IsCyclotomicExtension (insert n S) A B this : IsCyclotomicExtension S A ↥(adjoin A {b | ∃ n ∈ S, b ^ ↑n = 1}) ⊢ Module.Finite A B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S)
Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
case intro.refine'_2 n✝ : ℕ+ S✝ T : Set ℕ+ K : Type w L : Type z inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L h₁ : Finite ↑S✝ h✝ : Fintype ↑S✝ n : ℕ+ S : Set ℕ+ x✝¹ : n ∉ S x✝ : Set.Finite S H : ∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B] [h₂ : IsCyclotomicExtension S A B], Module.Finite A B A : Type u B : Type v inst✝³ : CommRing A inst✝² : CommRing B inst✝¹ : Algebra A B inst✝ : IsDomain B h : IsCyclotomicExtension (insert n S) A B this✝ : IsCyclotomicExtension S A ↥(adjoin A {b | ∃ n ∈ S, b ^ ↑n = 1}) this : Module.Finite A ↥(adjoin A {b | ∃ n ∈ S, b ^ ↑n = 1}) ⊢ Module.Finite A B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
n✝ : ℕ+ S✝ T : Set ℕ+ K : Type w L : Type z inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L h₁ : Finite ↑S✝ h✝ : Fintype ↑S✝ n : ℕ+ S : Set ℕ+ x✝¹ : n ∉ S x✝ : Set.Finite S H : ∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B] [h₂ : IsCyclotomicExtension S A B], Module.Finite A B A : Type u B : Type v inst✝³ : CommRing A inst✝² : CommRing B inst✝¹ : Algebra A B inst✝ : IsDomain B h : IsCyclotomicExtension (insert n S) A B this✝ : IsCyclotomicExtension S A ↥(adjoin A {b | ∃ n ∈ S, b ^ ↑n = 1}) this : Module.Finite A ↥(adjoin A {b | ∃ n ∈ S, b ^ ↑n = 1}) ⊢ Module.Finite (↥(adjoin A {b | ∃ n ∈ S, b ^ ↑n = 1})) B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
n✝ : ℕ+ S✝ T : Set ℕ+ K : Type w L : Type z inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L h₁ : Finite ↑S✝ h✝ : Fintype ↑S✝ n : ℕ+ S : Set ℕ+ x✝¹ : n ∉ S x✝ : Set.Finite S H : ∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B] [h₂ : IsCyclotomicExtension S A B], Module.Finite A B A : Type u B : Type v inst✝³ : CommRing A inst✝² : CommRing B inst✝¹ : Algebra A B inst✝ : IsDomain B h : IsCyclotomicExtension (S ∪ {n}) A B this✝ : IsCyclotomicExtension S A ↥(adjoin A {b | ∃ n ∈ S, b ^ ↑n = 1}) this : Module.Finite A ↥(adjoin A {b | ∃ n ∈ S, b ^ ↑n = 1}) ⊢ Module.Finite (↥(adjoin A {b | ∃ n ∈ S, b ^ ↑n = 1})) B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h
Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
n✝ : ℕ+ S✝ T : Set ℕ+ K : Type w L : Type z inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L h₁ : Finite ↑S✝ h✝ : Fintype ↑S✝ n : ℕ+ S : Set ℕ+ x✝¹ : n ∉ S x✝ : Set.Finite S H : ∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B] [h₂ : IsCyclotomicExtension S A B], Module.Finite A B A : Type u B : Type v inst✝³ : CommRing A inst✝² : CommRing B inst✝¹ : Algebra A B inst✝ : IsDomain B h : IsCyclotomicExtension (S ∪ {n}) A B this✝¹ : IsCyclotomicExtension S A ↥(adjoin A {b | ∃ n ∈ S, b ^ ↑n = 1}) this✝ : Module.Finite A ↥(adjoin A {b | ∃ n ∈ S, b ^ ↑n = 1}) this : IsCyclotomicExtension {n} (↥(adjoin A {b | ∃ a ∈ S, b ^ ↑a = 1})) B := union_right S {n} A B ⊢ Module.Finite (↥(adjoin A {b | ∃ n ∈ S, b ^ ↑n = 1})) B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h
Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
case intro.refine'_2 n✝ : ℕ+ S✝ T : Set ℕ+ K : Type w L : Type z inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L h₁ : Finite ↑S✝ h✝ : Fintype ↑S✝ n : ℕ+ S : Set ℕ+ x✝¹ : n ∉ S x✝ : Set.Finite S H : ∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B] [h₂ : IsCyclotomicExtension S A B], Module.Finite A B A : Type u B : Type v inst✝³ : CommRing A inst✝² : CommRing B inst✝¹ : Algebra A B inst✝ : IsDomain B h : IsCyclotomicExtension (insert n S) A B this✝¹ : IsCyclotomicExtension S A ↥(adjoin A {b | ∃ n ∈ S, b ^ ↑n = 1}) this✝ : Module.Finite A ↥(adjoin A {b | ∃ n ∈ S, b ^ ↑n = 1}) this : Module.Finite (↥(adjoin A {b | ∃ n ∈ S, b ^ ↑n = 1})) B ⊢ Module.Finite A B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _
Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁷ : CommRing A inst✝⁶ : CommRing B inst✝⁵ : Algebra A B inst✝⁴ : Field K inst✝³ : Field L inst✝² : Algebra K L h : NumberField K inst✝¹ : Finite ↑S inst✝ : IsCyclotomicExtension S K L ⊢ FiniteDimensional ℚ L
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
/-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by
Mathlib.NumberTheory.Cyclotomic.Basic.343_0.xReI1DeVvechFQU
/-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁷ : CommRing A inst✝⁶ : CommRing B inst✝⁵ : Algebra A B inst✝⁴ : Field K inst✝³ : Field L inst✝² : Algebra K L h : NumberField K inst✝¹ : Finite ↑S inst✝ : IsCyclotomicExtension S K L this : CharZero L ⊢ FiniteDimensional ℚ L
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
/-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
Mathlib.NumberTheory.Cyclotomic.Basic.343_0.xReI1DeVvechFQU
/-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁷ : CommRing A inst✝⁶ : CommRing B inst✝⁵ : Algebra A B inst✝⁴ : Field K inst✝³ : Field L inst✝² : Algebra K L h : NumberField K inst✝¹ : Finite ↑S inst✝ : IsCyclotomicExtension S K L this✝ : CharZero L this : Module.Finite K L ⊢ FiniteDimensional ℚ L
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _
/-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L
Mathlib.NumberTheory.Cyclotomic.Basic.343_0.xReI1DeVvechFQU
/-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L
Mathlib_NumberTheory_Cyclotomic_Basic
n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B ζ : B n : ℕ+ hζ : IsPrimitiveRoot ζ ↑n ⊢ adjoin A (rootSet (cyclotomic (↑n) A) B) = adjoin A {b | ∃ a ∈ {n}, b ^ ↑a = 1}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1}
Mathlib_NumberTheory_Cyclotomic_Basic
n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B ζ : B n : ℕ+ hζ : IsPrimitiveRoot ζ ↑n ⊢ adjoin A (rootSet (cyclotomic (↑n) A) B) = adjoin A {b | b ^ ↑n = 1}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1}
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_1 n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B ζ : B n : ℕ+ hζ : IsPrimitiveRoot ζ ↑n x : B hx : x ∈ rootSet (cyclotomic (↑n) A) B ⊢ x ∈ {b | b ^ ↑n = 1}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) ·
rw [mem_rootSet'] at hx
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) ·
Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1}
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_1 n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B ζ : B n : ℕ+ hζ : IsPrimitiveRoot ζ ↑n x : B hx : map (algebraMap A B) (cyclotomic (↑n) A) ≠ 0 ∧ (aeval x) (cyclotomic (↑n) A) = 0 ⊢ x ∈ {b | b ^ ↑n = 1}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx
Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1}
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_1 n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B ζ : B n : ℕ+ hζ : IsPrimitiveRoot ζ ↑n x : B hx : map (algebraMap A B) (cyclotomic (↑n) A) ≠ 0 ∧ (aeval x) (cyclotomic (↑n) A) = 0 ⊢ x ^ ↑n = 1
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1}
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_1 n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B ζ : B n : ℕ+ hζ : IsPrimitiveRoot ζ ↑n x : B hx : map (algebraMap A B) (cyclotomic (↑n) A) ≠ 0 ∧ (aeval x) (cyclotomic (↑n) A) = 0 ⊢ ∃ i ∈ Nat.divisors ↑n, IsRoot (cyclotomic i B) x
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos]
Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1}
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_1 n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B ζ : B n : ℕ+ hζ : IsPrimitiveRoot ζ ↑n x : B hx : map (algebraMap A B) (cyclotomic (↑n) A) ≠ 0 ∧ (aeval x) (cyclotomic (↑n) A) = 0 ⊢ IsRoot (cyclotomic (↑n) B) x
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1}
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_1 n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B ζ : B n : ℕ+ hζ : IsPrimitiveRoot ζ ↑n x : B hx : map (algebraMap A B) (cyclotomic (↑n) A) ≠ 0 ∧ (aeval x) (cyclotomic (↑n) A) = 0 ⊢ (aeval x) (cyclotomic (↑n) A) = 0
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1}
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_2 n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B ζ : B n : ℕ+ hζ : IsPrimitiveRoot ζ ↑n x : B hx : x ∈ {b | b ^ ↑n = 1} ⊢ x ∈ ↑(adjoin A (rootSet (cyclotomic (↑n) A) B))
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 ·
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 ·
Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1}
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_2 n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B ζ : B n : ℕ+ hζ : IsPrimitiveRoot ζ ↑n x : B hx : x ^ ↑n = 1 ⊢ x ∈ ↑(adjoin A (rootSet (cyclotomic (↑n) A) B))
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1}
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_2.intro.intro n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B ζ : B n : ℕ+ hζ : IsPrimitiveRoot ζ ↑n i : ℕ left✝ : i < ↑n hx : (ζ ^ i) ^ ↑n = 1 ⊢ ζ ^ i ∈ ↑(adjoin A (rootSet (cyclotomic (↑n) A) B))
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1}
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_2.intro.intro n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B ζ : B n : ℕ+ hζ : IsPrimitiveRoot ζ ↑n i : ℕ left✝ : i < ↑n hx : (ζ ^ i) ^ ↑n = 1 ⊢ ζ ∈ rootSet (cyclotomic (↑n) A) B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1}
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_2.intro.intro n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsDomain B ζ : B n : ℕ+ hζ : IsPrimitiveRoot ζ ↑n i : ℕ left✝ : i < ↑n hx : (ζ ^ i) ^ ↑n = 1 ⊢ cyclotomic (↑n) B ≠ 0 ∧ IsRoot (cyclotomic (↑n) B) ζ
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1}
Mathlib_NumberTheory_Cyclotomic_Basic
n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L n : ℕ+ inst✝ : IsDomain B ζ : B hζ : IsPrimitiveRoot ζ ↑n ⊢ adjoin A (rootSet (cyclotomic (↑n) A) B) = adjoin A {ζ}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
Mathlib.NumberTheory.Cyclotomic.Basic.389_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ}
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_1 n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L n : ℕ+ inst✝ : IsDomain B ζ : B hζ : IsPrimitiveRoot ζ ↑n x : B hx : x ∈ rootSet (cyclotomic (↑n) A) B ⊢ x ∈ ↑(adjoin A {ζ})
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) ·
suffices hx : x ^ n.1 = 1
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) ·
Mathlib.NumberTheory.Cyclotomic.Basic.389_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ}
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_1 n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L n : ℕ+ inst✝ : IsDomain B ζ : B hζ : IsPrimitiveRoot ζ ↑n x : B hx✝ : x ∈ rootSet (cyclotomic (↑n) A) B hx : x ^ ↑n = 1 ⊢ x ∈ ↑(adjoin A {ζ}) case hx n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L n : ℕ+ inst✝ : IsDomain B ζ : B hζ : IsPrimitiveRoot ζ ↑n x : B hx : x ∈ rootSet (cyclotomic (↑n) A) B ⊢ x ^ ↑n = 1
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1
Mathlib.NumberTheory.Cyclotomic.Basic.389_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ}
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_1.intro.intro n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L n : ℕ+ inst✝ : IsDomain B ζ : B hζ : IsPrimitiveRoot ζ ↑n i : ℕ left✝ : i < ↑n hx✝ : ζ ^ i ∈ rootSet (cyclotomic (↑n) A) B hx : (ζ ^ i) ^ ↑n = 1 ⊢ ζ ^ i ∈ ↑(adjoin A {ζ}) case hx n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L n : ℕ+ inst✝ : IsDomain B ζ : B hζ : IsPrimitiveRoot ζ ↑n x : B hx : x ∈ rootSet (cyclotomic (↑n) A) B ⊢ x ^ ↑n = 1
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
Mathlib.NumberTheory.Cyclotomic.Basic.389_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ}
Mathlib_NumberTheory_Cyclotomic_Basic
case hx n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L n : ℕ+ inst✝ : IsDomain B ζ : B hζ : IsPrimitiveRoot ζ ↑n x : B hx : x ∈ rootSet (cyclotomic (↑n) A) B ⊢ x ^ ↑n = 1
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
Mathlib.NumberTheory.Cyclotomic.Basic.389_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ}
Mathlib_NumberTheory_Cyclotomic_Basic
case hx n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L n : ℕ+ inst✝ : IsDomain B ζ : B hζ : IsPrimitiveRoot ζ ↑n x : B hx : x ∈ rootSet (cyclotomic (↑n) A) B ⊢ ∃ i ∈ Nat.divisors ↑n, IsRoot (cyclotomic i B) x
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _
Mathlib.NumberTheory.Cyclotomic.Basic.389_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ}
Mathlib_NumberTheory_Cyclotomic_Basic
case hx n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L n : ℕ+ inst✝ : IsDomain B ζ : B hζ : IsPrimitiveRoot ζ ↑n x : B hx : x ∈ rootSet (cyclotomic (↑n) A) B ⊢ IsRoot (cyclotomic (↑n) B) x
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
Mathlib.NumberTheory.Cyclotomic.Basic.389_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ}
Mathlib_NumberTheory_Cyclotomic_Basic
case hx n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L n : ℕ+ inst✝ : IsDomain B ζ : B hζ : IsPrimitiveRoot ζ ↑n x : B hx : cyclotomic (↑n) B ≠ 0 ∧ IsRoot (cyclotomic (↑n) B) x ⊢ IsRoot (cyclotomic (↑n) B) x
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
Mathlib.NumberTheory.Cyclotomic.Basic.389_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ}
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_2 n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L n : ℕ+ inst✝ : IsDomain B ζ : B hζ : IsPrimitiveRoot ζ ↑n x : B hx : x ∈ {ζ} ⊢ x ∈ rootSet (cyclotomic (↑n) A) B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 ·
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 ·
Mathlib.NumberTheory.Cyclotomic.Basic.389_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ}
Mathlib_NumberTheory_Cyclotomic_Basic
case refine'_2 n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L n : ℕ+ inst✝ : IsDomain B ζ : B hζ : IsPrimitiveRoot ζ ↑n x : B hx : x = ζ ⊢ x ∈ rootSet (cyclotomic (↑n) A) B
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
Mathlib.NumberTheory.Cyclotomic.Basic.389_0.xReI1DeVvechFQU
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ}
Mathlib_NumberTheory_Cyclotomic_Basic
n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L n : ℕ+ inst✝ : IsDomain B h : IsCyclotomicExtension {n} A B ζ : B hζ : IsPrimitiveRoot ζ ↑n ⊢ adjoin A {ζ} = ⊤
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
Mathlib.NumberTheory.Cyclotomic.Basic.404_0.xReI1DeVvechFQU
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤
Mathlib_NumberTheory_Cyclotomic_Basic
n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L n : ℕ+ inst✝ : IsDomain B h : IsCyclotomicExtension {n} A B ζ : B hζ : IsPrimitiveRoot ζ ↑n ⊢ adjoin A {ζ} = ⊤
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical
Mathlib.NumberTheory.Cyclotomic.Basic.404_0.xReI1DeVvechFQU
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤
Mathlib_NumberTheory_Cyclotomic_Basic
n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L n : ℕ+ inst✝ : IsDomain B h : IsCyclotomicExtension {n} A B ζ : B hζ : IsPrimitiveRoot ζ ↑n ⊢ adjoin A (rootSet (cyclotomic (↑n) A) B) = ⊤
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
Mathlib.NumberTheory.Cyclotomic.Basic.404_0.xReI1DeVvechFQU
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤
Mathlib_NumberTheory_Cyclotomic_Basic
n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L n : ℕ+ inst✝ : IsDomain B h : IsCyclotomicExtension {n} A B ζ : B hζ : IsPrimitiveRoot ζ ↑n ⊢ adjoin A {b | ∃ a ∈ {n}, b ^ ↑a = 1} = ⊤
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
Mathlib.NumberTheory.Cyclotomic.Basic.404_0.xReI1DeVvechFQU
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤
Mathlib_NumberTheory_Cyclotomic_Basic
n✝¹ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L ζ : B n : ℕ+ h : IsPrimitiveRoot ζ ↑n n✝ : ℕ+ hi : n✝ ∈ {n} ⊢ ∃ r, IsPrimitiveRoot r ↑n✝
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by
Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B))
Mathlib_NumberTheory_Cyclotomic_Basic
n✝¹ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L ζ : B n : ℕ+ h : IsPrimitiveRoot ζ ↑n n✝ : ℕ+ hi : n✝ = n ⊢ ∃ r, IsPrimitiveRoot r ↑n✝
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi
Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B))
Mathlib_NumberTheory_Cyclotomic_Basic
n✝¹ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L ζ : B n : ℕ+ h : IsPrimitiveRoot ζ ↑n n✝ : ℕ+ hi : n✝ = n ⊢ IsPrimitiveRoot { val := ζ, property := (_ : ζ ∈ ↑(adjoin A {ζ})) } ↑n✝
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B))
Mathlib_NumberTheory_Cyclotomic_Basic
n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L ζ : B n : ℕ+ h : IsPrimitiveRoot ζ ↑n x : ↥(adjoin A {ζ}) ⊢ x ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by
refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by
Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B))
Mathlib_NumberTheory_Cyclotomic_Basic
case refine_1 n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L ζ : B n : ℕ+ h : IsPrimitiveRoot ζ ↑n x : ↥(adjoin A {ζ}) b : B hb : b ∈ {ζ} ⊢ { val := b, property := (_ : b ∈ ↑(adjoin A {ζ})) } ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) ·
rw [Set.mem_singleton_iff] at hb
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) ·
Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B))
Mathlib_NumberTheory_Cyclotomic_Basic
case refine_1 n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L ζ : B n : ℕ+ h : IsPrimitiveRoot ζ ↑n x : ↥(adjoin A {ζ}) b : B hb✝ : b ∈ {ζ} hb : b = ζ ⊢ { val := b, property := (_ : b ∈ ↑(adjoin A {ζ})) } ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb
Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B))
Mathlib_NumberTheory_Cyclotomic_Basic
case refine_1 n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L ζ : B n : ℕ+ h : IsPrimitiveRoot ζ ↑n x : ↥(adjoin A {ζ}) b : B hb✝ : b ∈ {ζ} hb : b = ζ ⊢ { val := b, property := (_ : b ∈ ↑(adjoin A {ζ})) } ∈ {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _
Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B))
Mathlib_NumberTheory_Cyclotomic_Basic
case refine_1 n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L ζ : B n : ℕ+ h : IsPrimitiveRoot ζ ↑n x : ↥(adjoin A {ζ}) b : B hb✝ : b ∈ {ζ} hb : b = ζ ⊢ { val := ζ, property := (_ : (fun x => x ∈ adjoin A {ζ}) ζ) } ^ ↑n = 1
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B))
Mathlib_NumberTheory_Cyclotomic_Basic
case refine_1 n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L ζ : B n : ℕ+ h : IsPrimitiveRoot ζ ↑n x : ↥(adjoin A {ζ}) b : B hb✝ : b ∈ {ζ} hb : b = ζ ⊢ ζ ^ ↑n = 1
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B))
Mathlib_NumberTheory_Cyclotomic_Basic
case refine_2 n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L ζ : B n : ℕ+ h : IsPrimitiveRoot ζ ↑n x : ↥(adjoin A {ζ}) a : A ⊢ (algebraMap A ↥(adjoin A {ζ})) a ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 ·
exact Subalgebra.algebraMap_mem _ _
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 ·
Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B))
Mathlib_NumberTheory_Cyclotomic_Basic
case refine_3 n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L ζ : B n : ℕ+ h : IsPrimitiveRoot ζ ↑n x b₁ b₂ : ↥(adjoin A {ζ}) hb₁ : b₁ ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} hb₂ : b₂ ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} ⊢ b₁ + b₂ ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 · exact Subalgebra.algebraMap_mem _ _ ·
exact Subalgebra.add_mem _ hb₁ hb₂
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 · exact Subalgebra.algebraMap_mem _ _ ·
Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B))
Mathlib_NumberTheory_Cyclotomic_Basic
case refine_4 n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L ζ : B n : ℕ+ h : IsPrimitiveRoot ζ ↑n x b₁ b₂ : ↥(adjoin A {ζ}) hb₁ : b₁ ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} hb₂ : b₂ ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} ⊢ b₁ * b₂ ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 · exact Subalgebra.algebraMap_mem _ _ · exact Subalgebra.add_mem _ hb₁ hb₂ ·
exact Subalgebra.mul_mem _ hb₁ hb₂
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 · exact Subalgebra.algebraMap_mem _ _ · exact Subalgebra.add_mem _ hb₁ hb₂ ·
Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B))
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L H : IsCyclotomicExtension S K L hS : n ∈ S ⊢ Splits (algebraMap K L) (X ^ ↑n - 1)
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 · exact Subalgebra.algebraMap_mem _ _ · exact Subalgebra.add_mem _ hb₁ hb₂ · exact Subalgebra.mul_mem _ hb₁ hb₂ } #align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension end section Field variable {n S} /-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/ theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow, Polynomial.map_X]
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/ theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
Mathlib.NumberTheory.Cyclotomic.Basic.441_0.xReI1DeVvechFQU
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/ theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (X ^ (n : ℕ) - 1)
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L H : IsCyclotomicExtension S K L hS : n ∈ S ⊢ Splits (RingHom.id L) (X ^ ↑n - 1)
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 · exact Subalgebra.algebraMap_mem _ _ · exact Subalgebra.add_mem _ hb₁ hb₂ · exact Subalgebra.mul_mem _ hb₁ hb₂ } #align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension end section Field variable {n S} /-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/ theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow, Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/ theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow, Polynomial.map_X]
Mathlib.NumberTheory.Cyclotomic.Basic.441_0.xReI1DeVvechFQU
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/ theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (X ^ (n : ℕ) - 1)
Mathlib_NumberTheory_Cyclotomic_Basic
case intro n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁵ : CommRing A inst✝⁴ : CommRing B inst✝³ : Algebra A B inst✝² : Field K inst✝¹ : Field L inst✝ : Algebra K L H : IsCyclotomicExtension S K L hS : n ∈ S z : L hz : IsPrimitiveRoot z ↑n ⊢ Splits (RingHom.id L) (X ^ ↑n - 1)
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 · exact Subalgebra.algebraMap_mem _ _ · exact Subalgebra.add_mem _ hb₁ hb₂ · exact Subalgebra.mul_mem _ hb₁ hb₂ } #align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension end section Field variable {n S} /-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/ theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow, Polynomial.map_X] obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/ theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow, Polynomial.map_X] obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
Mathlib.NumberTheory.Cyclotomic.Basic.441_0.xReI1DeVvechFQU
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/ theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (X ^ (n : ℕ) - 1)
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsCyclotomicExtension S K L hS : n ∈ S ⊢ Splits (algebraMap K L) (cyclotomic (↑n) K)
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 · exact Subalgebra.algebraMap_mem _ _ · exact Subalgebra.add_mem _ hb₁ hb₂ · exact Subalgebra.mul_mem _ hb₁ hb₂ } #align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension end section Field variable {n S} /-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/ theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow, Polynomial.map_X] obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS exact X_pow_sub_one_splits hz set_option linter.uppercaseLean3 false in #align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one /-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/ theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/ theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (cyclotomic n K) := by
Mathlib.NumberTheory.Cyclotomic.Basic.451_0.xReI1DeVvechFQU
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/ theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (cyclotomic n K)
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsCyclotomicExtension S K L hS : n ∈ S ⊢ cyclotomic (↑n) K ∣ X ^ ↑n - C 1
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 · exact Subalgebra.algebraMap_mem _ _ · exact Subalgebra.add_mem _ hb₁ hb₂ · exact Subalgebra.mul_mem _ hb₁ hb₂ } #align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension end section Field variable {n S} /-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/ theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow, Polynomial.map_X] obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS exact X_pow_sub_one_splits hz set_option linter.uppercaseLean3 false in #align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one /-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/ theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (cyclotomic n K) := by refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/ theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (cyclotomic n K) := by refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
Mathlib.NumberTheory.Cyclotomic.Basic.451_0.xReI1DeVvechFQU
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/ theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (cyclotomic n K)
Mathlib_NumberTheory_Cyclotomic_Basic
case h n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsCyclotomicExtension S K L hS : n ∈ S ⊢ X ^ ↑n - C 1 = cyclotomic (↑n) K * ∏ i in Nat.properDivisors ↑n, cyclotomic i K
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 · exact Subalgebra.algebraMap_mem _ _ · exact Subalgebra.add_mem _ hb₁ hb₂ · exact Subalgebra.mul_mem _ hb₁ hb₂ } #align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension end section Field variable {n S} /-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/ theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow, Polynomial.map_X] obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS exact X_pow_sub_one_splits hz set_option linter.uppercaseLean3 false in #align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one /-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/ theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (cyclotomic n K) := by refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _ use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/ theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (cyclotomic n K) := by refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _ use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
Mathlib.NumberTheory.Cyclotomic.Basic.451_0.xReI1DeVvechFQU
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/ theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (cyclotomic n K)
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsCyclotomicExtension {n} K L ⊢ adjoin K (rootSet (X ^ ↑n - 1) L) = ⊤
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 · exact Subalgebra.algebraMap_mem _ _ · exact Subalgebra.add_mem _ hb₁ hb₂ · exact Subalgebra.mul_mem _ hb₁ hb₂ } #align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension end section Field variable {n S} /-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/ theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow, Polynomial.map_X] obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS exact X_pow_sub_one_splits hz set_option linter.uppercaseLean3 false in #align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one /-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/ theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (cyclotomic n K) := by refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _ use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one] #align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic variable (n S) section Singleton variable [IsCyclotomicExtension {n} K L] /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/ theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) := { splits' := splits_X_pow_sub_one K L (mem_singleton n) adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/ theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) := { splits' := splits_X_pow_sub_one K L (mem_singleton n) adjoin_rootSet' := by
Mathlib.NumberTheory.Cyclotomic.Basic.465_0.xReI1DeVvechFQU
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/ theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1)
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsCyclotomicExtension {n} K L ⊢ adjoin K (rootSet (X ^ ↑n - 1) L) = adjoin K {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 · exact Subalgebra.algebraMap_mem _ _ · exact Subalgebra.add_mem _ hb₁ hb₂ · exact Subalgebra.mul_mem _ hb₁ hb₂ } #align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension end section Field variable {n S} /-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/ theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow, Polynomial.map_X] obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS exact X_pow_sub_one_splits hz set_option linter.uppercaseLean3 false in #align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one /-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/ theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (cyclotomic n K) := by refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _ use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one] #align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic variable (n S) section Singleton variable [IsCyclotomicExtension {n} K L] /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/ theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) := { splits' := splits_X_pow_sub_one K L (mem_singleton n) adjoin_rootSet' := by rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/ theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) := { splits' := splits_X_pow_sub_one K L (mem_singleton n) adjoin_rootSet' := by rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
Mathlib.NumberTheory.Cyclotomic.Basic.465_0.xReI1DeVvechFQU
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/ theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1)
Mathlib_NumberTheory_Cyclotomic_Basic
case e_s n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsCyclotomicExtension {n} K L ⊢ rootSet (X ^ ↑n - 1) L = {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 · exact Subalgebra.algebraMap_mem _ _ · exact Subalgebra.add_mem _ hb₁ hb₂ · exact Subalgebra.mul_mem _ hb₁ hb₂ } #align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension end section Field variable {n S} /-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/ theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow, Polynomial.map_X] obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS exact X_pow_sub_one_splits hz set_option linter.uppercaseLean3 false in #align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one /-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/ theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (cyclotomic n K) := by refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _ use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one] #align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic variable (n S) section Singleton variable [IsCyclotomicExtension {n} K L] /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/ theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) := { splits' := splits_X_pow_sub_one K L (mem_singleton n) adjoin_rootSet' := by rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2] congr
refine' Set.ext fun x => _
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/ theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) := { splits' := splits_X_pow_sub_one K L (mem_singleton n) adjoin_rootSet' := by rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2] congr
Mathlib.NumberTheory.Cyclotomic.Basic.465_0.xReI1DeVvechFQU
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/ theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1)
Mathlib_NumberTheory_Cyclotomic_Basic
case e_s n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsCyclotomicExtension {n} K L x : L ⊢ x ∈ rootSet (X ^ ↑n - 1) L ↔ x ∈ {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 · exact Subalgebra.algebraMap_mem _ _ · exact Subalgebra.add_mem _ hb₁ hb₂ · exact Subalgebra.mul_mem _ hb₁ hb₂ } #align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension end section Field variable {n S} /-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/ theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow, Polynomial.map_X] obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS exact X_pow_sub_one_splits hz set_option linter.uppercaseLean3 false in #align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one /-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/ theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (cyclotomic n K) := by refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _ use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one] #align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic variable (n S) section Singleton variable [IsCyclotomicExtension {n} K L] /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/ theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) := { splits' := splits_X_pow_sub_one K L (mem_singleton n) adjoin_rootSet' := by rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2] congr refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left, mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/ theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) := { splits' := splits_X_pow_sub_one K L (mem_singleton n) adjoin_rootSet' := by rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2] congr refine' Set.ext fun x => _
Mathlib.NumberTheory.Cyclotomic.Basic.465_0.xReI1DeVvechFQU
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/ theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1)
Mathlib_NumberTheory_Cyclotomic_Basic
case e_s n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsCyclotomicExtension {n} K L x : L ⊢ x ∈ rootSet (X ^ ↑n - 1) L ↔ x ^ ↑n = 1
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 · exact Subalgebra.algebraMap_mem _ _ · exact Subalgebra.add_mem _ hb₁ hb₂ · exact Subalgebra.mul_mem _ hb₁ hb₂ } #align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension end section Field variable {n S} /-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/ theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow, Polynomial.map_X] obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS exact X_pow_sub_one_splits hz set_option linter.uppercaseLean3 false in #align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one /-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/ theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (cyclotomic n K) := by refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _ use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one] #align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic variable (n S) section Singleton variable [IsCyclotomicExtension {n} K L] /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/ theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) := { splits' := splits_X_pow_sub_one K L (mem_singleton n) adjoin_rootSet' := by rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2] congr refine' Set.ext fun x => _ simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left, mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X, and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/ theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) := { splits' := splits_X_pow_sub_one K L (mem_singleton n) adjoin_rootSet' := by rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2] congr refine' Set.ext fun x => _ simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left, mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
Mathlib.NumberTheory.Cyclotomic.Basic.465_0.xReI1DeVvechFQU
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/ theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1)
Mathlib_NumberTheory_Cyclotomic_Basic
case e_s n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsCyclotomicExtension {n} K L x : L ⊢ x ^ ↑n = 1 → X ^ ↑n - 1 ≠ 0
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 · exact Subalgebra.algebraMap_mem _ _ · exact Subalgebra.add_mem _ hb₁ hb₂ · exact Subalgebra.mul_mem _ hb₁ hb₂ } #align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension end section Field variable {n S} /-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/ theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow, Polynomial.map_X] obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS exact X_pow_sub_one_splits hz set_option linter.uppercaseLean3 false in #align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one /-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/ theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (cyclotomic n K) := by refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _ use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one] #align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic variable (n S) section Singleton variable [IsCyclotomicExtension {n} K L] /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/ theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) := { splits' := splits_X_pow_sub_one K L (mem_singleton n) adjoin_rootSet' := by rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2] congr refine' Set.ext fun x => _ simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left, mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub] simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X, and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L)
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/ theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) := { splits' := splits_X_pow_sub_one K L (mem_singleton n) adjoin_rootSet' := by rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2] congr refine' Set.ext fun x => _ simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left, mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub] simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X, and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
Mathlib.NumberTheory.Cyclotomic.Basic.465_0.xReI1DeVvechFQU
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/ theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1)
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsCyclotomicExtension {n} K L ⊢ adjoin K (rootSet (cyclotomic (↑n) K) L) = ⊤
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 · exact Subalgebra.algebraMap_mem _ _ · exact Subalgebra.add_mem _ hb₁ hb₂ · exact Subalgebra.mul_mem _ hb₁ hb₂ } #align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension end section Field variable {n S} /-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/ theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow, Polynomial.map_X] obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS exact X_pow_sub_one_splits hz set_option linter.uppercaseLean3 false in #align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one /-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/ theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (cyclotomic n K) := by refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _ use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one] #align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic variable (n S) section Singleton variable [IsCyclotomicExtension {n} K L] /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/ theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) := { splits' := splits_X_pow_sub_one K L (mem_singleton n) adjoin_rootSet' := by rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2] congr refine' Set.ext fun x => _ simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left, mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub] simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X, and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one] exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) } set_option linter.uppercaseLean3 false in #align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one /-- Any two `n`-th cyclotomic extensions are isomorphic. -/ def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] : L ≃ₐ[K] L' := let h₁ := isSplittingField_X_pow_sub_one n K L let h₂ := isSplittingField_X_pow_sub_one n K L' (@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans (@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm #align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one theorem isGalois : IsGalois K L := letI := isSplittingField_X_pow_sub_one n K L IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2 (IsCyclotomicExtension.neZero' n K L).1) #align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/ theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) := { splits' := splits_cyclotomic K L (mem_singleton n) adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/ theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) := { splits' := splits_cyclotomic K L (mem_singleton n) adjoin_rootSet' := by
Mathlib.NumberTheory.Cyclotomic.Basic.497_0.xReI1DeVvechFQU
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/ theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K)
Mathlib_NumberTheory_Cyclotomic_Basic
n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁶ : CommRing A inst✝⁵ : CommRing B inst✝⁴ : Algebra A B inst✝³ : Field K inst✝² : Field L inst✝¹ : Algebra K L inst✝ : IsCyclotomicExtension {n} K L ⊢ adjoin K (rootSet (cyclotomic (↑n) K) L) = adjoin K {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1}
/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.NumberTheory.NumberField.Basic import Mathlib.FieldTheory.Galois #align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba" /-! # Cyclotomic extensions Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. ## Main definitions * `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all `n ∈ S`. * `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. * `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n` is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. ## Main results * `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and `IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if `Function.Injective (algebraMap B C)`. * `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then `IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`. * `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then `IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`. * `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. * `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a number field. * `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. * `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. ## Implementation details Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains. All results are in the `IsCyclotomicExtension` namespace. Note that some results, for example `IsCyclotomicExtension.trans`, `IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`, `IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and `CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are included in the `Cyclotomic` locale. -/ open Polynomial Algebra FiniteDimensional Set open scoped BigOperators universe u v w z variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z) variable [CommRing A] [CommRing B] [Algebra A B] variable [Field K] [Field L] [Algebra K L] noncomputable section /-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated over `A` by the roots of `X ^ n - 1`. -/ @[mk_iff] class IsCyclotomicExtension : Prop where /-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/ exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n /-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/ adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} #align is_cyclotomic_extension IsCyclotomicExtension namespace IsCyclotomicExtension section Basic /-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/ theorem iff_adjoin_eq_top : IsCyclotomicExtension S A B ↔ (∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ := ⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h => ⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩ #align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top /-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/ theorem iff_singleton : IsCyclotomicExtension {n} A B ↔ (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by simp [IsCyclotomicExtension_iff] #align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton /-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h #align is_cyclotomic_extension.empty IsCyclotomicExtension.empty /-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/ theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ := Algebra.eq_top_iff.2 fun x => by simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x #align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ A B := by -- Porting note: Lean3 is able to infer `A`. refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩ rw [← h] at hx simpa using hx #align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top variable (A B) /-- Transitivity of cyclotomic extensions. -/ theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C] (h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by refine' ⟨fun hn => _, fun x => _⟩ · cases' hn with hn hn · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn refine' ⟨algebraMap B C b, _⟩ exact hb.map_of_injective h · exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn · refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x) (fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy · let f := IsScalarTower.toAlgHom A B C have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f := ⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩ rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb refine' adjoin_mono (fun y hy => _) hb obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩ #align is_cyclotomic_extension.trans IsCyclotomicExtension.trans @[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq] intro t ht obtain ⟨ζ, hζ⟩ := hprim ht rw [mem_singleton_iff, ← PNat.coe_eq_one_iff] exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ) · rintro (rfl | rfl) -- Porting note: `R := A` was not needed. · exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ · rw [iff_singleton] exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩ #align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B` is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by roots of unity of order in `T`. -/ theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] : IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} = {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪ {b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by refine' le_antisymm _ _ · rintro x ⟨n, hn₁ | hn₂, hnpow⟩ · left; exact ⟨n, hn₁, hnpow⟩ · right; exact ⟨n, hn₂, hnpow⟩ · rintro x (⟨n, hn⟩ | ⟨n, hn⟩) · exact ⟨n, Or.inl hn.1, hn.2⟩ · exact ⟨n, Or.inr hn.1, hn.2⟩ refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩ replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h #align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right /-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`, then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B` given by roots of unity of order in `S`. -/ theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) : IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by refine' ⟨@fun n hn => _, fun b => _⟩ · obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] · convert mem_top (R := A) (x := b) rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] norm_cast #align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left variable {n S} /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] : IsCyclotomicExtension (S ∪ {n}) A B := by refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩ · rw [mem_union, mem_singleton_iff] at hs obtain hs | rfl := hs · exact H.exists_prim_root hs · obtain ⟨m, hm⟩ := hS obtain ⟨x, rfl⟩ := h m hm obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm refine' ⟨ζ ^ (x : ℕ), _⟩ convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s) simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos] · refine' _root_.eq_top_iff.2 _ rw [← ((iff_adjoin_eq_top S A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, hm⟩ := hx exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩ #align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd /-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if `IsCyclotomicExtension (S ∪ {n}) A B`. -/ theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by refine' ⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩ · exact H.exists_prim_root (subset_union_left _ _ hs) · rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2] refine' adjoin_mono fun x hx => _ simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢ obtain ⟨m, rfl | hm, hxpow⟩ := hx · obtain ⟨y, hy⟩ := hS refine' ⟨y, ⟨hy, _⟩⟩ obtain ⟨z, rfl⟩ := h y hy simp only [PNat.mul_coe, pow_mul, hxpow, one_pow] · exact ⟨m, ⟨hm, hxpow⟩⟩ #align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd variable (n S) /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/ theorem iff_union_singleton_one : IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by obtain hS | rfl := S.eq_empty_or_nonempty.symm · exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS rw [empty_union] refine' ⟨fun H => _, fun H => _⟩ · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ simp [adjoin_singleton_one, empty] · refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ simp [@singleton_one A B _ _ _ H] #align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one variable {A B} /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension {1} A B := by convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) simp #align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top /-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/ theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) : IsCyclotomicExtension {1} A B := singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm #align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective variable (A B) /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then `IsCyclotomicExtension S A C`. -/ protected theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B] (f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) #align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv protected theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h exact hr.neZero' #align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero protected theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by haveI := IsCyclotomicExtension.neZero n A B exact NeZero.nat_of_neZero (algebraMap A B) #align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero' end Basic section Fintype theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] : Module.Finite A B := by classical rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] refine' fg_adjoin_of_finite _ fun b hb => _ · simp only [mem_singleton_iff, exists_eq_left] have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset := Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ rw [this] exact (nthRoots (↑n) 1).toFinset.finite_toSet · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ #align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/ protected theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] : Module.Finite A B := by cases' nonempty_fintype S with h revert h₂ A B refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ · intro _ _ _ _ _ refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] · intro _ _ _ _ h haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) := union_left _ (insert n S) _ _ (subset_insert n S) haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by rw [← union_singleton] at h letI := @union_right S {n} A B _ _ _ h exact finite_of_singleton n _ _ exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ #align is_cyclotomic_extension.finite IsCyclotomicExtension.finite /-- A cyclotomic finite extension of a number field is a number field. -/ theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L := { to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective to_finiteDimensional := by haveI := charZero_of_injective_algebraMap (algebraMap K L).injective haveI := IsCyclotomicExtension.finite S K L exact Module.Finite.trans K _ } #align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField /-- A finite cyclotomic extension of an integral noetherian domain is integral -/ theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] : Algebra.IsIntegral A B := letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance #align is_cyclotomic_extension.integral IsCyclotomicExtension.integral /-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/ theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C] [IsCyclotomicExtension S K C] : FiniteDimensional K C := IsCyclotomicExtension.finite S K C #align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional end Fintype section variable {A B} theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) · rw [mem_rootSet'] at hx simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] rw [isRoot_of_unity_iff n.pos] refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) · suffices hx : x ^ n.1 = 1 obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) refine' (isRoot_of_unity_iff n.pos B).2 _ refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx exact hx.2 · simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) #align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B] {ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by classical rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] exact ((iff_adjoin_eq_top {n} A B).mp h).2 #align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top variable (A) theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+} (h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) := { exists_prim_root := fun hi => by rw [Set.mem_singleton_iff] at hi refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] adjoin_roots := fun x => by refine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_) · rw [Set.mem_singleton_iff] at hb refine' subset_adjoin _ simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 · exact Subalgebra.algebraMap_mem _ _ · exact Subalgebra.add_mem _ hb₁ hb₂ · exact Subalgebra.mul_mem _ hb₁ hb₂ } #align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension end section Field variable {n S} /-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/ theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow, Polynomial.map_X] obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS exact X_pow_sub_one_splits hz set_option linter.uppercaseLean3 false in #align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one /-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/ theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) : Splits (algebraMap K L) (cyclotomic n K) := by refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _ use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one] #align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic variable (n S) section Singleton variable [IsCyclotomicExtension {n} K L] /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/ theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) := { splits' := splits_X_pow_sub_one K L (mem_singleton n) adjoin_rootSet' := by rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2] congr refine' Set.ext fun x => _ simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left, mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub] simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X, and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one] exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) } set_option linter.uppercaseLean3 false in #align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one /-- Any two `n`-th cyclotomic extensions are isomorphic. -/ def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] : L ≃ₐ[K] L' := let h₁ := isSplittingField_X_pow_sub_one n K L let h₂ := isSplittingField_X_pow_sub_one n K L' (@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans (@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm #align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one theorem isGalois : IsGalois K L := letI := isSplittingField_X_pow_sub_one n K L IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2 (IsCyclotomicExtension.neZero' n K L).1) #align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/ theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) := { splits' := splits_cyclotomic K L (mem_singleton n) adjoin_rootSet' := by rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/ theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) := { splits' := splits_cyclotomic K L (mem_singleton n) adjoin_rootSet' := by rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
Mathlib.NumberTheory.Cyclotomic.Basic.497_0.xReI1DeVvechFQU
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/ theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K)
Mathlib_NumberTheory_Cyclotomic_Basic