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L3 Lab 3

F : Type u_1 α✝ : Type u_2 β : Type u_3 γ : Type u_4 x✝¹ : NonAssocSemiring α✝ x✝ : NonAssocSemiring β α : Type u_5 inst✝ : NonAssocSemiring α ⊢ α →+* α

/- Copyright (c) 2019 Amelia Livingston. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Amelia Livingston, Jireh Loreaux -/ import Mathlib.Algebra.Ring.Defs import Mathlib.Algebra.Ring.Basic import Mathlib.Data.Pi.Algebra #align_import algebra.hom.ring from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9" /-! # Homomorphisms of semirings and rings This file defines bundled homomorphisms of (non-unital) semirings and rings. As with monoid and groups, we use the same structure `RingHom a β`, a.k.a. `α →+* β`, for both types of homomorphisms. ## Main definitions * `NonUnitalRingHom`: Non-unital (semi)ring homomorphisms. Additive monoid homomorphism which preserve multiplication. * `RingHom`: (Semi)ring homomorphisms. Monoid homomorphisms which are also additive monoid homomorphism. ## Notations * `→ₙ+*`: Non-unital (semi)ring homs * `→+*`: (Semi)ring homs ## Implementation notes * There's a coercion from bundled homs to fun, and the canonical notation is to use the bundled hom as a function via this coercion. * There is no `SemiringHom` -- the idea is that `RingHom` is used. The constructor for a `RingHom` between semirings needs a proof of `map_zero`, `map_one` and `map_add` as well as `map_mul`; a separate constructor `RingHom.mk'` will construct ring homs between rings from monoid homs given only a proof that addition is preserved. ## Tags `RingHom`, `SemiringHom` -/ open Function variable {F α β γ : Type*} /-- Bundled non-unital semiring homomorphisms `α →ₙ+* β`; use this for bundled non-unital ring homomorphisms too. When possible, instead of parametrizing results over `(f : α →ₙ+* β)`, you should parametrize over `(F : Type*) [NonUnitalRingHomClass F α β] (f : F)`. When you extend this structure, make sure to extend `NonUnitalRingHomClass`. -/ structure NonUnitalRingHom (α β : Type*) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends α →ₙ* β, α →+ β #align non_unital_ring_hom NonUnitalRingHom /-- `α →ₙ+* β` denotes the type of non-unital ring homomorphisms from `α` to `β`. -/ infixr:25 " →ₙ+* " => NonUnitalRingHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as a semigroup homomorphism `α →ₙ* β`. The `simp`-normal form is `(f : α →ₙ* β)`. -/ add_decl_doc NonUnitalRingHom.toMulHom #align non_unital_ring_hom.to_mul_hom NonUnitalRingHom.toMulHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc NonUnitalRingHom.toAddMonoidHom #align non_unital_ring_hom.to_add_monoid_hom NonUnitalRingHom.toAddMonoidHom section NonUnitalRingHomClass /-- `NonUnitalRingHomClass F α β` states that `F` is a type of non-unital (semi)ring homomorphisms. You should extend this class when you extend `NonUnitalRingHom`. -/ class NonUnitalRingHomClass (F : Type*) (α β : outParam (Type*)) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends MulHomClass F α β, AddMonoidHomClass F α β #align non_unital_ring_hom_class NonUnitalRingHomClass variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] [NonUnitalRingHomClass F α β] /-- Turn an element of a type `F` satisfying `NonUnitalRingHomClass F α β` into an actual `NonUnitalRingHom`. This is declared as the default coercion from `F` to `α →ₙ+* β`. -/ @[coe] def NonUnitalRingHomClass.toNonUnitalRingHom (f : F) : α →ₙ+* β := { (f : α →ₙ* β), (f : α →+ β) with } /-- Any type satisfying `NonUnitalRingHomClass` can be cast into `NonUnitalRingHom` via `NonUnitalRingHomClass.toNonUnitalRingHom`. -/ instance : CoeTC F (α →ₙ+* β) := ⟨NonUnitalRingHomClass.toNonUnitalRingHom⟩ end NonUnitalRingHomClass namespace NonUnitalRingHom section coe variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] instance : NonUnitalRingHomClass (α →ₙ+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := NonUnitalRingHom.map_add' map_zero := NonUnitalRingHom.map_zero' map_mul f := f.map_mul' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -/ -- instance : CoeFun (α →ₙ+* β) fun _ => α → β := -- ⟨fun f => f.toFun⟩ -- Porting note: removed due to new `coe` in Lean4 #noalign non_unital_ring_hom.to_fun_eq_coe #noalign non_unital_ring_hom.coe_mk #noalign non_unital_ring_hom.coe_coe initialize_simps_projections NonUnitalRingHom (toFun → apply) @[simp] theorem coe_toMulHom (f : α →ₙ+* β) : ⇑f.toMulHom = f := rfl #align non_unital_ring_hom.coe_to_mul_hom NonUnitalRingHom.coe_toMulHom @[simp] theorem coe_mulHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →ₙ* β) = ⟨f, h₁⟩ := rfl #align non_unital_ring_hom.coe_mul_hom_mk NonUnitalRingHom.coe_mulHom_mk theorem coe_toAddMonoidHom (f : α →ₙ+* β) : ⇑f.toAddMonoidHom = f := rfl #align non_unital_ring_hom.coe_to_add_monoid_hom NonUnitalRingHom.coe_toAddMonoidHom @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →+ β) = ⟨⟨f, h₂⟩, h₃⟩ := rfl #align non_unital_ring_hom.coe_add_monoid_hom_mk NonUnitalRingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : α →ₙ+* β := { f.toMulHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align non_unital_ring_hom.copy NonUnitalRingHom.copy @[simp] theorem coe_copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align non_unital_ring_hom.coe_copy NonUnitalRingHom.coe_copy theorem copy_eq (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align non_unital_ring_hom.copy_eq NonUnitalRingHom.copy_eq end coe section variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] variable (f : α →ₙ+* β) {x y : α} @[ext] theorem ext ⦃f g : α →ₙ+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align non_unital_ring_hom.ext NonUnitalRingHom.ext theorem ext_iff {f g : α →ₙ+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align non_unital_ring_hom.ext_iff NonUnitalRingHom.ext_iff @[simp] theorem mk_coe (f : α →ₙ+* β) (h₁ h₂ h₃) : NonUnitalRingHom.mk (MulHom.mk f h₁) h₂ h₃ = f := ext fun _ => rfl #align non_unital_ring_hom.mk_coe NonUnitalRingHom.mk_coe theorem coe_addMonoidHom_injective : Injective fun f : α →ₙ+* β => (f : α →+ β) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align non_unital_ring_hom.coe_add_monoid_hom_injective NonUnitalRingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_mulHom_injective : Injective fun f : α →ₙ+* β => (f : α →ₙ* β) := fun _ _ h => ext <| MulHom.congr_fun h #align non_unital_ring_hom.coe_mul_hom_injective NonUnitalRingHom.coe_mulHom_injective end variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/ protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by refine' { toFun := id.. } <;> intros <;> rfl #align non_unital_ring_hom.id NonUnitalRingHom.id instance : Zero (α →ₙ+* β) := ⟨{ toFun := 0, map_mul' := fun _ _ => (mul_zero (0 : β)).symm, map_zero' := rfl, map_add' := fun _ _ => (add_zero (0 : β)).symm }⟩ instance : Inhabited (α →ₙ+* β) := ⟨0⟩ @[simp] theorem coe_zero : ⇑(0 : α →ₙ+* β) = 0 := rfl #align non_unital_ring_hom.coe_zero NonUnitalRingHom.coe_zero @[simp] theorem zero_apply (x : α) : (0 : α →ₙ+* β) x = 0 := rfl #align non_unital_ring_hom.zero_apply NonUnitalRingHom.zero_apply @[simp] theorem id_apply (x : α) : NonUnitalRingHom.id α x = x := rfl #align non_unital_ring_hom.id_apply NonUnitalRingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (NonUnitalRingHom.id α : α →+ α) = AddMonoidHom.id α := rfl #align non_unital_ring_hom.coe_add_monoid_hom_id NonUnitalRingHom.coe_addMonoidHom_id @[simp] theorem coe_mulHom_id : (NonUnitalRingHom.id α : α →ₙ* α) = MulHom.id α := rfl #align non_unital_ring_hom.coe_mul_hom_id NonUnitalRingHom.coe_mulHom_id variable [NonUnitalNonAssocSemiring γ] /-- Composition of non-unital ring homomorphisms is a non-unital ring homomorphism. -/ def comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : α →ₙ+* γ := { g.toMulHom.comp f.toMulHom, g.toAddMonoidHom.comp f.toAddMonoidHom with } #align non_unital_ring_hom.comp NonUnitalRingHom.comp /-- Composition of non-unital ring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonUnitalNonAssocSemiring δ} (f : α →ₙ+* β) (g : β →ₙ+* γ) (h : γ →ₙ+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align non_unital_ring_hom.comp_assoc NonUnitalRingHom.comp_assoc @[simp] theorem coe_comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : ⇑(g.comp f) = g ∘ f := rfl #align non_unital_ring_hom.coe_comp NonUnitalRingHom.coe_comp @[simp] theorem comp_apply (g : β →ₙ+* γ) (f : α →ₙ+* β) (x : α) : g.comp f x = g (f x) := rfl #align non_unital_ring_hom.comp_apply NonUnitalRingHom.comp_apply variable (g : β →ₙ+* γ) (f : α →ₙ+* β) @[simp] theorem coe_comp_addMonoidHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : AddMonoidHom.mk ⟨g ∘ f, (g.comp f).map_zero'⟩ (g.comp f).map_add' = (g : β →+ γ).comp f := rfl #align non_unital_ring_hom.coe_comp_add_monoid_hom NonUnitalRingHom.coe_comp_addMonoidHom @[simp] theorem coe_comp_mulHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : MulHom.mk (g ∘ f) (g.comp f).map_mul' = (g : β →ₙ* γ).comp f := rfl #align non_unital_ring_hom.coe_comp_mul_hom NonUnitalRingHom.coe_comp_mulHom @[simp] theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 := by ext simp #align non_unital_ring_hom.comp_zero NonUnitalRingHom.comp_zero @[simp] theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 := by ext rfl #align non_unital_ring_hom.zero_comp NonUnitalRingHom.zero_comp @[simp] theorem comp_id (f : α →ₙ+* β) : f.comp (NonUnitalRingHom.id α) = f := ext fun _ => rfl #align non_unital_ring_hom.comp_id NonUnitalRingHom.comp_id @[simp] theorem id_comp (f : α →ₙ+* β) : (NonUnitalRingHom.id β).comp f = f := ext fun _ => rfl #align non_unital_ring_hom.id_comp NonUnitalRingHom.id_comp instance : MonoidWithZero (α →ₙ+* α) where one := NonUnitalRingHom.id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ zero := 0 mul_zero := comp_zero zero_mul := zero_comp theorem one_def : (1 : α →ₙ+* α) = NonUnitalRingHom.id α := rfl #align non_unital_ring_hom.one_def NonUnitalRingHom.one_def @[simp] theorem coe_one : ⇑(1 : α →ₙ+* α) = id := rfl #align non_unital_ring_hom.coe_one NonUnitalRingHom.coe_one theorem mul_def (f g : α →ₙ+* α) : f * g = f.comp g := rfl #align non_unital_ring_hom.mul_def NonUnitalRingHom.mul_def @[simp] theorem coe_mul (f g : α →ₙ+* α) : ⇑(f * g) = f ∘ g := rfl #align non_unital_ring_hom.coe_mul NonUnitalRingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →ₙ+* γ} {f : α →ₙ+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_right NonUnitalRingHom.cancel_right @[simp] theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_left NonUnitalRingHom.cancel_left end NonUnitalRingHom /-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too. This extends from both `MonoidHom` and `MonoidWithZeroHom` in order to put the fields in a sensible order, even though `MonoidWithZeroHom` already extends `MonoidHom`. -/ structure RingHom (α : Type*) (β : Type*) [NonAssocSemiring α] [NonAssocSemiring β] extends α →* β, α →+ β, α →ₙ+* β, α →*₀ β #align ring_hom RingHom /-- `α →+* β` denotes the type of ring homomorphisms from `α` to `β`. -/ infixr:25 " →+* " => RingHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid with zero homomorphism `α →*₀ β`. The `simp`-normal form is `(f : α →*₀ β)`. -/ add_decl_doc RingHom.toMonoidWithZeroHom #align ring_hom.to_monoid_with_zero_hom RingHom.toMonoidWithZeroHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid homomorphism `α →* β`. The `simp`-normal form is `(f : α →* β)`. -/ add_decl_doc RingHom.toMonoidHom #align ring_hom.to_monoid_hom RingHom.toMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc RingHom.toAddMonoidHom #align ring_hom.to_add_monoid_hom RingHom.toAddMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a non-unital ring homomorphism `α →ₙ+* β`. The `simp`-normal form is `(f : α →ₙ+* β)`. -/ add_decl_doc RingHom.toNonUnitalRingHom #align ring_hom.to_non_unital_ring_hom RingHom.toNonUnitalRingHom section RingHomClass /-- `RingHomClass F α β` states that `F` is a type of (semi)ring homomorphisms. You should extend this class when you extend `RingHom`. This extends from both `MonoidHomClass` and `MonoidWithZeroHomClass` in order to put the fields in a sensible order, even though `MonoidWithZeroHomClass` already extends `MonoidHomClass`. -/ class RingHomClass (F : Type*) (α β : outParam (Type*)) [NonAssocSemiring α] [NonAssocSemiring β] extends MonoidHomClass F α β, AddMonoidHomClass F α β, MonoidWithZeroHomClass F α β #align ring_hom_class RingHomClass set_option linter.deprecated false in /-- Ring homomorphisms preserve `bit1`. -/ @[simp] lemma map_bit1 [NonAssocSemiring α] [NonAssocSemiring β] [RingHomClass F α β] (f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) := by simp [bit1] #align map_bit1 map_bit1 -- Porting note: marked `{}` rather than `[]` to prevent dangerous instances variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} [RingHomClass F α β] /-- Turn an element of a type `F` satisfying `RingHomClass F α β` into an actual `RingHom`. This is declared as the default coercion from `F` to `α →+* β`. -/ @[coe] def RingHomClass.toRingHom (f : F) : α →+* β := { (f : α →* β), (f : α →+ β) with } /-- Any type satisfying `RingHomClass` can be cast into `RingHom` via `RingHomClass.toRingHom`. -/ instance : CoeTC F (α →+* β) := ⟨RingHomClass.toRingHom⟩ instance (priority := 100) RingHomClass.toNonUnitalRingHomClass : NonUnitalRingHomClass F α β := { ‹RingHomClass F α β› with } #align ring_hom_class.to_non_unital_ring_hom_class RingHomClass.toNonUnitalRingHomClass end RingHomClass namespace RingHom section coe /-! Throughout this section, some `Semiring` arguments are specified with `{}` instead of `[]`. See note [implicit instance arguments]. -/ variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} instance instRingHomClass : RingHomClass (α →+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := RingHom.map_add' map_zero := RingHom.map_zero' map_mul f := f.map_mul' map_one f := f.map_one' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -- -/ -- instance : CoeFun (α →+* β) fun _ => α → β := -- ⟨RingHom.toFun⟩ initialize_simps_projections RingHom (toFun → apply) -- Porting note: is this lemma still needed in Lean4? -- Porting note: because `f.toFun` really means `f.toMonoidHom.toOneHom.toFun` and -- `toMonoidHom_eq_coe` wants to simplify `f.toMonoidHom` to `(↑f : M →* N)`, this can't -- be a simp lemma anymore -- @[simp] theorem toFun_eq_coe (f : α →+* β) : f.toFun = f := rfl #align ring_hom.to_fun_eq_coe RingHom.toFun_eq_coe @[simp] theorem coe_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α → β) = f := rfl #align ring_hom.coe_mk RingHom.coe_mk @[simp] theorem coe_coe {F : Type*} [RingHomClass F α β] (f : F) : ((f : α →+* β) : α → β) = f := rfl #align ring_hom.coe_coe RingHom.coe_coe attribute [coe] RingHom.toMonoidHom instance coeToMonoidHom : Coe (α →+* β) (α →* β) := ⟨RingHom.toMonoidHom⟩ #align ring_hom.has_coe_monoid_hom RingHom.coeToMonoidHom -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_monoid_hom @[simp] theorem toMonoidHom_eq_coe (f : α →+* β) : f.toMonoidHom = f := rfl #align ring_hom.to_monoid_hom_eq_coe RingHom.toMonoidHom_eq_coe -- Porting note: this can't be a simp lemma anymore -- @[simp] theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f := by rfl #align ring_hom.to_monoid_with_zero_hom_eq_coe RingHom.toMonoidWithZeroHom_eq_coe @[simp] theorem coe_monoidHom_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α →* β) = f := rfl #align ring_hom.coe_monoid_hom_mk RingHom.coe_monoidHom_mk -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_add_monoid_hom @[simp] theorem toAddMonoidHom_eq_coe (f : α →+* β) : f.toAddMonoidHom = f := rfl #align ring_hom.to_add_monoid_hom_eq_coe RingHom.toAddMonoidHom_eq_coe @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃ h₄) : ((⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨⟨f, h₃⟩, h₄⟩ := rfl #align ring_hom.coe_add_monoid_hom_mk RingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ def copy (f : α →+* β) (f' : α → β) (h : f' = f) : α →+* β := { f.toMonoidWithZeroHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align ring_hom.copy RingHom.copy @[simp] theorem coe_copy (f : α →+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align ring_hom.coe_copy RingHom.coe_copy theorem copy_eq (f : α →+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align ring_hom.copy_eq RingHom.copy_eq end coe section variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} (f : α →+* β) {x y : α} theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x := FunLike.congr_fun h x #align ring_hom.congr_fun RingHom.congr_fun theorem congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y := FunLike.congr_arg f h #align ring_hom.congr_arg RingHom.congr_arg theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g := FunLike.coe_injective h #align ring_hom.coe_inj RingHom.coe_inj @[ext] theorem ext ⦃f g : α →+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align ring_hom.ext RingHom.ext theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align ring_hom.ext_iff RingHom.ext_iff @[simp] theorem mk_coe (f : α →+* β) (h₁ h₂ h₃ h₄) : RingHom.mk ⟨⟨f, h₁⟩, h₂⟩ h₃ h₄ = f := ext fun _ => rfl #align ring_hom.mk_coe RingHom.mk_coe theorem coe_addMonoidHom_injective : Injective (fun f : α →+* β => (f : α →+ β)) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align ring_hom.coe_add_monoid_hom_injective RingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_monoidHom_injective : Injective (fun f : α →+* β => (f : α →* β)) := fun _ _ h => ext <| MonoidHom.congr_fun h #align ring_hom.coe_monoid_hom_injective RingHom.coe_monoidHom_injective /-- Ring homomorphisms map zero to zero. -/ protected theorem map_zero (f : α →+* β) : f 0 = 0 := map_zero f #align ring_hom.map_zero RingHom.map_zero /-- Ring homomorphisms map one to one. -/ protected theorem map_one (f : α →+* β) : f 1 = 1 := map_one f #align ring_hom.map_one RingHom.map_one /-- Ring homomorphisms preserve addition. -/ protected theorem map_add (f : α →+* β) : ∀ a b, f (a + b) = f a + f b := map_add f #align ring_hom.map_add RingHom.map_add /-- Ring homomorphisms preserve multiplication. -/ protected theorem map_mul (f : α →+* β) : ∀ a b, f (a * b) = f a * f b := map_mul f #align ring_hom.map_mul RingHom.map_mul @[simp] theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 0 1) = ite p 0 1 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_zero_one RingHom.map_ite_zero_one @[simp] theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 1 0) = ite p 1 0 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_one_zero RingHom.map_ite_one_zero /-- `f : α →+* β` has a trivial codomain iff `f 1 = 0`. -/ theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 := by rw [map_one, eq_comm] #align ring_hom.codomain_trivial_iff_map_one_eq_zero RingHom.codomain_trivial_iff_map_one_eq_zero /-- `f : α →+* β` has a trivial codomain iff it has a trivial range. -/ theorem codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 := f.codomain_trivial_iff_map_one_eq_zero.trans ⟨fun h x => by rw [← mul_one x, map_mul, h, mul_zero], fun h => h 1⟩ #align ring_hom.codomain_trivial_iff_range_trivial RingHom.codomain_trivial_iff_range_trivial /-- `f : α →+* β` doesn't map `1` to `0` if `β` is nontrivial -/ theorem map_one_ne_zero [Nontrivial β] : f 1 ≠ 0 := mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one #align ring_hom.map_one_ne_zero RingHom.map_one_ne_zero /-- If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. -/ theorem domain_nontrivial [Nontrivial β] : Nontrivial α := ⟨⟨1, 0, mt (fun h => show f 1 = 0 by rw [h, map_zero]) f.map_one_ne_zero⟩⟩ #align ring_hom.domain_nontrivial RingHom.domain_nontrivial theorem codomain_trivial (f : α →+* β) [h : Subsingleton α] : Subsingleton β := (subsingleton_or_nontrivial β).resolve_right fun _ => not_nontrivial_iff_subsingleton.mpr h f.domain_nontrivial #align ring_hom.codomain_trivial RingHom.codomain_trivial end /-- Ring homomorphisms preserve additive inverse. -/ protected theorem map_neg [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x : α) : f (-x) = -f x := map_neg f x #align ring_hom.map_neg RingHom.map_neg /-- Ring homomorphisms preserve subtraction. -/ protected theorem map_sub [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x y : α) : f (x - y) = f x - f y := map_sub f x y #align ring_hom.map_sub RingHom.map_sub /-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/ def mk' [NonAssocSemiring α] [NonAssocRing β] (f : α →* β) (map_add : ∀ a b, f (a + b) = f a + f b) : α →+* β := { AddMonoidHom.mk' f map_add, f with } #align ring_hom.mk' RingHom.mk' variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} /-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by

refine' { toFun := _root_.id.. }

/-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by

Mathlib.Algebra.Ring.Hom.Defs.631_0.KyHvVYrIs9pW9ZQ

/-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α

Mathlib_Algebra_Ring_Hom_Defs

case refine'_1 F : Type u_1 α✝ : Type u_2 β : Type u_3 γ : Type u_4 x✝¹ : NonAssocSemiring α✝ x✝ : NonAssocSemiring β α : Type u_5 inst✝ : NonAssocSemiring α ⊢ _root_.id 1 = 1

/- Copyright (c) 2019 Amelia Livingston. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Amelia Livingston, Jireh Loreaux -/ import Mathlib.Algebra.Ring.Defs import Mathlib.Algebra.Ring.Basic import Mathlib.Data.Pi.Algebra #align_import algebra.hom.ring from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9" /-! # Homomorphisms of semirings and rings This file defines bundled homomorphisms of (non-unital) semirings and rings. As with monoid and groups, we use the same structure `RingHom a β`, a.k.a. `α →+* β`, for both types of homomorphisms. ## Main definitions * `NonUnitalRingHom`: Non-unital (semi)ring homomorphisms. Additive monoid homomorphism which preserve multiplication. * `RingHom`: (Semi)ring homomorphisms. Monoid homomorphisms which are also additive monoid homomorphism. ## Notations * `→ₙ+*`: Non-unital (semi)ring homs * `→+*`: (Semi)ring homs ## Implementation notes * There's a coercion from bundled homs to fun, and the canonical notation is to use the bundled hom as a function via this coercion. * There is no `SemiringHom` -- the idea is that `RingHom` is used. The constructor for a `RingHom` between semirings needs a proof of `map_zero`, `map_one` and `map_add` as well as `map_mul`; a separate constructor `RingHom.mk'` will construct ring homs between rings from monoid homs given only a proof that addition is preserved. ## Tags `RingHom`, `SemiringHom` -/ open Function variable {F α β γ : Type*} /-- Bundled non-unital semiring homomorphisms `α →ₙ+* β`; use this for bundled non-unital ring homomorphisms too. When possible, instead of parametrizing results over `(f : α →ₙ+* β)`, you should parametrize over `(F : Type*) [NonUnitalRingHomClass F α β] (f : F)`. When you extend this structure, make sure to extend `NonUnitalRingHomClass`. -/ structure NonUnitalRingHom (α β : Type*) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends α →ₙ* β, α →+ β #align non_unital_ring_hom NonUnitalRingHom /-- `α →ₙ+* β` denotes the type of non-unital ring homomorphisms from `α` to `β`. -/ infixr:25 " →ₙ+* " => NonUnitalRingHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as a semigroup homomorphism `α →ₙ* β`. The `simp`-normal form is `(f : α →ₙ* β)`. -/ add_decl_doc NonUnitalRingHom.toMulHom #align non_unital_ring_hom.to_mul_hom NonUnitalRingHom.toMulHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc NonUnitalRingHom.toAddMonoidHom #align non_unital_ring_hom.to_add_monoid_hom NonUnitalRingHom.toAddMonoidHom section NonUnitalRingHomClass /-- `NonUnitalRingHomClass F α β` states that `F` is a type of non-unital (semi)ring homomorphisms. You should extend this class when you extend `NonUnitalRingHom`. -/ class NonUnitalRingHomClass (F : Type*) (α β : outParam (Type*)) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends MulHomClass F α β, AddMonoidHomClass F α β #align non_unital_ring_hom_class NonUnitalRingHomClass variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] [NonUnitalRingHomClass F α β] /-- Turn an element of a type `F` satisfying `NonUnitalRingHomClass F α β` into an actual `NonUnitalRingHom`. This is declared as the default coercion from `F` to `α →ₙ+* β`. -/ @[coe] def NonUnitalRingHomClass.toNonUnitalRingHom (f : F) : α →ₙ+* β := { (f : α →ₙ* β), (f : α →+ β) with } /-- Any type satisfying `NonUnitalRingHomClass` can be cast into `NonUnitalRingHom` via `NonUnitalRingHomClass.toNonUnitalRingHom`. -/ instance : CoeTC F (α →ₙ+* β) := ⟨NonUnitalRingHomClass.toNonUnitalRingHom⟩ end NonUnitalRingHomClass namespace NonUnitalRingHom section coe variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] instance : NonUnitalRingHomClass (α →ₙ+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := NonUnitalRingHom.map_add' map_zero := NonUnitalRingHom.map_zero' map_mul f := f.map_mul' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -/ -- instance : CoeFun (α →ₙ+* β) fun _ => α → β := -- ⟨fun f => f.toFun⟩ -- Porting note: removed due to new `coe` in Lean4 #noalign non_unital_ring_hom.to_fun_eq_coe #noalign non_unital_ring_hom.coe_mk #noalign non_unital_ring_hom.coe_coe initialize_simps_projections NonUnitalRingHom (toFun → apply) @[simp] theorem coe_toMulHom (f : α →ₙ+* β) : ⇑f.toMulHom = f := rfl #align non_unital_ring_hom.coe_to_mul_hom NonUnitalRingHom.coe_toMulHom @[simp] theorem coe_mulHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →ₙ* β) = ⟨f, h₁⟩ := rfl #align non_unital_ring_hom.coe_mul_hom_mk NonUnitalRingHom.coe_mulHom_mk theorem coe_toAddMonoidHom (f : α →ₙ+* β) : ⇑f.toAddMonoidHom = f := rfl #align non_unital_ring_hom.coe_to_add_monoid_hom NonUnitalRingHom.coe_toAddMonoidHom @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →+ β) = ⟨⟨f, h₂⟩, h₃⟩ := rfl #align non_unital_ring_hom.coe_add_monoid_hom_mk NonUnitalRingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : α →ₙ+* β := { f.toMulHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align non_unital_ring_hom.copy NonUnitalRingHom.copy @[simp] theorem coe_copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align non_unital_ring_hom.coe_copy NonUnitalRingHom.coe_copy theorem copy_eq (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align non_unital_ring_hom.copy_eq NonUnitalRingHom.copy_eq end coe section variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] variable (f : α →ₙ+* β) {x y : α} @[ext] theorem ext ⦃f g : α →ₙ+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align non_unital_ring_hom.ext NonUnitalRingHom.ext theorem ext_iff {f g : α →ₙ+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align non_unital_ring_hom.ext_iff NonUnitalRingHom.ext_iff @[simp] theorem mk_coe (f : α →ₙ+* β) (h₁ h₂ h₃) : NonUnitalRingHom.mk (MulHom.mk f h₁) h₂ h₃ = f := ext fun _ => rfl #align non_unital_ring_hom.mk_coe NonUnitalRingHom.mk_coe theorem coe_addMonoidHom_injective : Injective fun f : α →ₙ+* β => (f : α →+ β) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align non_unital_ring_hom.coe_add_monoid_hom_injective NonUnitalRingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_mulHom_injective : Injective fun f : α →ₙ+* β => (f : α →ₙ* β) := fun _ _ h => ext <| MulHom.congr_fun h #align non_unital_ring_hom.coe_mul_hom_injective NonUnitalRingHom.coe_mulHom_injective end variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/ protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by refine' { toFun := id.. } <;> intros <;> rfl #align non_unital_ring_hom.id NonUnitalRingHom.id instance : Zero (α →ₙ+* β) := ⟨{ toFun := 0, map_mul' := fun _ _ => (mul_zero (0 : β)).symm, map_zero' := rfl, map_add' := fun _ _ => (add_zero (0 : β)).symm }⟩ instance : Inhabited (α →ₙ+* β) := ⟨0⟩ @[simp] theorem coe_zero : ⇑(0 : α →ₙ+* β) = 0 := rfl #align non_unital_ring_hom.coe_zero NonUnitalRingHom.coe_zero @[simp] theorem zero_apply (x : α) : (0 : α →ₙ+* β) x = 0 := rfl #align non_unital_ring_hom.zero_apply NonUnitalRingHom.zero_apply @[simp] theorem id_apply (x : α) : NonUnitalRingHom.id α x = x := rfl #align non_unital_ring_hom.id_apply NonUnitalRingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (NonUnitalRingHom.id α : α →+ α) = AddMonoidHom.id α := rfl #align non_unital_ring_hom.coe_add_monoid_hom_id NonUnitalRingHom.coe_addMonoidHom_id @[simp] theorem coe_mulHom_id : (NonUnitalRingHom.id α : α →ₙ* α) = MulHom.id α := rfl #align non_unital_ring_hom.coe_mul_hom_id NonUnitalRingHom.coe_mulHom_id variable [NonUnitalNonAssocSemiring γ] /-- Composition of non-unital ring homomorphisms is a non-unital ring homomorphism. -/ def comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : α →ₙ+* γ := { g.toMulHom.comp f.toMulHom, g.toAddMonoidHom.comp f.toAddMonoidHom with } #align non_unital_ring_hom.comp NonUnitalRingHom.comp /-- Composition of non-unital ring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonUnitalNonAssocSemiring δ} (f : α →ₙ+* β) (g : β →ₙ+* γ) (h : γ →ₙ+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align non_unital_ring_hom.comp_assoc NonUnitalRingHom.comp_assoc @[simp] theorem coe_comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : ⇑(g.comp f) = g ∘ f := rfl #align non_unital_ring_hom.coe_comp NonUnitalRingHom.coe_comp @[simp] theorem comp_apply (g : β →ₙ+* γ) (f : α →ₙ+* β) (x : α) : g.comp f x = g (f x) := rfl #align non_unital_ring_hom.comp_apply NonUnitalRingHom.comp_apply variable (g : β →ₙ+* γ) (f : α →ₙ+* β) @[simp] theorem coe_comp_addMonoidHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : AddMonoidHom.mk ⟨g ∘ f, (g.comp f).map_zero'⟩ (g.comp f).map_add' = (g : β →+ γ).comp f := rfl #align non_unital_ring_hom.coe_comp_add_monoid_hom NonUnitalRingHom.coe_comp_addMonoidHom @[simp] theorem coe_comp_mulHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : MulHom.mk (g ∘ f) (g.comp f).map_mul' = (g : β →ₙ* γ).comp f := rfl #align non_unital_ring_hom.coe_comp_mul_hom NonUnitalRingHom.coe_comp_mulHom @[simp] theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 := by ext simp #align non_unital_ring_hom.comp_zero NonUnitalRingHom.comp_zero @[simp] theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 := by ext rfl #align non_unital_ring_hom.zero_comp NonUnitalRingHom.zero_comp @[simp] theorem comp_id (f : α →ₙ+* β) : f.comp (NonUnitalRingHom.id α) = f := ext fun _ => rfl #align non_unital_ring_hom.comp_id NonUnitalRingHom.comp_id @[simp] theorem id_comp (f : α →ₙ+* β) : (NonUnitalRingHom.id β).comp f = f := ext fun _ => rfl #align non_unital_ring_hom.id_comp NonUnitalRingHom.id_comp instance : MonoidWithZero (α →ₙ+* α) where one := NonUnitalRingHom.id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ zero := 0 mul_zero := comp_zero zero_mul := zero_comp theorem one_def : (1 : α →ₙ+* α) = NonUnitalRingHom.id α := rfl #align non_unital_ring_hom.one_def NonUnitalRingHom.one_def @[simp] theorem coe_one : ⇑(1 : α →ₙ+* α) = id := rfl #align non_unital_ring_hom.coe_one NonUnitalRingHom.coe_one theorem mul_def (f g : α →ₙ+* α) : f * g = f.comp g := rfl #align non_unital_ring_hom.mul_def NonUnitalRingHom.mul_def @[simp] theorem coe_mul (f g : α →ₙ+* α) : ⇑(f * g) = f ∘ g := rfl #align non_unital_ring_hom.coe_mul NonUnitalRingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →ₙ+* γ} {f : α →ₙ+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_right NonUnitalRingHom.cancel_right @[simp] theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_left NonUnitalRingHom.cancel_left end NonUnitalRingHom /-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too. This extends from both `MonoidHom` and `MonoidWithZeroHom` in order to put the fields in a sensible order, even though `MonoidWithZeroHom` already extends `MonoidHom`. -/ structure RingHom (α : Type*) (β : Type*) [NonAssocSemiring α] [NonAssocSemiring β] extends α →* β, α →+ β, α →ₙ+* β, α →*₀ β #align ring_hom RingHom /-- `α →+* β` denotes the type of ring homomorphisms from `α` to `β`. -/ infixr:25 " →+* " => RingHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid with zero homomorphism `α →*₀ β`. The `simp`-normal form is `(f : α →*₀ β)`. -/ add_decl_doc RingHom.toMonoidWithZeroHom #align ring_hom.to_monoid_with_zero_hom RingHom.toMonoidWithZeroHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid homomorphism `α →* β`. The `simp`-normal form is `(f : α →* β)`. -/ add_decl_doc RingHom.toMonoidHom #align ring_hom.to_monoid_hom RingHom.toMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc RingHom.toAddMonoidHom #align ring_hom.to_add_monoid_hom RingHom.toAddMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a non-unital ring homomorphism `α →ₙ+* β`. The `simp`-normal form is `(f : α →ₙ+* β)`. -/ add_decl_doc RingHom.toNonUnitalRingHom #align ring_hom.to_non_unital_ring_hom RingHom.toNonUnitalRingHom section RingHomClass /-- `RingHomClass F α β` states that `F` is a type of (semi)ring homomorphisms. You should extend this class when you extend `RingHom`. This extends from both `MonoidHomClass` and `MonoidWithZeroHomClass` in order to put the fields in a sensible order, even though `MonoidWithZeroHomClass` already extends `MonoidHomClass`. -/ class RingHomClass (F : Type*) (α β : outParam (Type*)) [NonAssocSemiring α] [NonAssocSemiring β] extends MonoidHomClass F α β, AddMonoidHomClass F α β, MonoidWithZeroHomClass F α β #align ring_hom_class RingHomClass set_option linter.deprecated false in /-- Ring homomorphisms preserve `bit1`. -/ @[simp] lemma map_bit1 [NonAssocSemiring α] [NonAssocSemiring β] [RingHomClass F α β] (f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) := by simp [bit1] #align map_bit1 map_bit1 -- Porting note: marked `{}` rather than `[]` to prevent dangerous instances variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} [RingHomClass F α β] /-- Turn an element of a type `F` satisfying `RingHomClass F α β` into an actual `RingHom`. This is declared as the default coercion from `F` to `α →+* β`. -/ @[coe] def RingHomClass.toRingHom (f : F) : α →+* β := { (f : α →* β), (f : α →+ β) with } /-- Any type satisfying `RingHomClass` can be cast into `RingHom` via `RingHomClass.toRingHom`. -/ instance : CoeTC F (α →+* β) := ⟨RingHomClass.toRingHom⟩ instance (priority := 100) RingHomClass.toNonUnitalRingHomClass : NonUnitalRingHomClass F α β := { ‹RingHomClass F α β› with } #align ring_hom_class.to_non_unital_ring_hom_class RingHomClass.toNonUnitalRingHomClass end RingHomClass namespace RingHom section coe /-! Throughout this section, some `Semiring` arguments are specified with `{}` instead of `[]`. See note [implicit instance arguments]. -/ variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} instance instRingHomClass : RingHomClass (α →+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := RingHom.map_add' map_zero := RingHom.map_zero' map_mul f := f.map_mul' map_one f := f.map_one' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -- -/ -- instance : CoeFun (α →+* β) fun _ => α → β := -- ⟨RingHom.toFun⟩ initialize_simps_projections RingHom (toFun → apply) -- Porting note: is this lemma still needed in Lean4? -- Porting note: because `f.toFun` really means `f.toMonoidHom.toOneHom.toFun` and -- `toMonoidHom_eq_coe` wants to simplify `f.toMonoidHom` to `(↑f : M →* N)`, this can't -- be a simp lemma anymore -- @[simp] theorem toFun_eq_coe (f : α →+* β) : f.toFun = f := rfl #align ring_hom.to_fun_eq_coe RingHom.toFun_eq_coe @[simp] theorem coe_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α → β) = f := rfl #align ring_hom.coe_mk RingHom.coe_mk @[simp] theorem coe_coe {F : Type*} [RingHomClass F α β] (f : F) : ((f : α →+* β) : α → β) = f := rfl #align ring_hom.coe_coe RingHom.coe_coe attribute [coe] RingHom.toMonoidHom instance coeToMonoidHom : Coe (α →+* β) (α →* β) := ⟨RingHom.toMonoidHom⟩ #align ring_hom.has_coe_monoid_hom RingHom.coeToMonoidHom -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_monoid_hom @[simp] theorem toMonoidHom_eq_coe (f : α →+* β) : f.toMonoidHom = f := rfl #align ring_hom.to_monoid_hom_eq_coe RingHom.toMonoidHom_eq_coe -- Porting note: this can't be a simp lemma anymore -- @[simp] theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f := by rfl #align ring_hom.to_monoid_with_zero_hom_eq_coe RingHom.toMonoidWithZeroHom_eq_coe @[simp] theorem coe_monoidHom_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α →* β) = f := rfl #align ring_hom.coe_monoid_hom_mk RingHom.coe_monoidHom_mk -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_add_monoid_hom @[simp] theorem toAddMonoidHom_eq_coe (f : α →+* β) : f.toAddMonoidHom = f := rfl #align ring_hom.to_add_monoid_hom_eq_coe RingHom.toAddMonoidHom_eq_coe @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃ h₄) : ((⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨⟨f, h₃⟩, h₄⟩ := rfl #align ring_hom.coe_add_monoid_hom_mk RingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ def copy (f : α →+* β) (f' : α → β) (h : f' = f) : α →+* β := { f.toMonoidWithZeroHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align ring_hom.copy RingHom.copy @[simp] theorem coe_copy (f : α →+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align ring_hom.coe_copy RingHom.coe_copy theorem copy_eq (f : α →+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align ring_hom.copy_eq RingHom.copy_eq end coe section variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} (f : α →+* β) {x y : α} theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x := FunLike.congr_fun h x #align ring_hom.congr_fun RingHom.congr_fun theorem congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y := FunLike.congr_arg f h #align ring_hom.congr_arg RingHom.congr_arg theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g := FunLike.coe_injective h #align ring_hom.coe_inj RingHom.coe_inj @[ext] theorem ext ⦃f g : α →+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align ring_hom.ext RingHom.ext theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align ring_hom.ext_iff RingHom.ext_iff @[simp] theorem mk_coe (f : α →+* β) (h₁ h₂ h₃ h₄) : RingHom.mk ⟨⟨f, h₁⟩, h₂⟩ h₃ h₄ = f := ext fun _ => rfl #align ring_hom.mk_coe RingHom.mk_coe theorem coe_addMonoidHom_injective : Injective (fun f : α →+* β => (f : α →+ β)) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align ring_hom.coe_add_monoid_hom_injective RingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_monoidHom_injective : Injective (fun f : α →+* β => (f : α →* β)) := fun _ _ h => ext <| MonoidHom.congr_fun h #align ring_hom.coe_monoid_hom_injective RingHom.coe_monoidHom_injective /-- Ring homomorphisms map zero to zero. -/ protected theorem map_zero (f : α →+* β) : f 0 = 0 := map_zero f #align ring_hom.map_zero RingHom.map_zero /-- Ring homomorphisms map one to one. -/ protected theorem map_one (f : α →+* β) : f 1 = 1 := map_one f #align ring_hom.map_one RingHom.map_one /-- Ring homomorphisms preserve addition. -/ protected theorem map_add (f : α →+* β) : ∀ a b, f (a + b) = f a + f b := map_add f #align ring_hom.map_add RingHom.map_add /-- Ring homomorphisms preserve multiplication. -/ protected theorem map_mul (f : α →+* β) : ∀ a b, f (a * b) = f a * f b := map_mul f #align ring_hom.map_mul RingHom.map_mul @[simp] theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 0 1) = ite p 0 1 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_zero_one RingHom.map_ite_zero_one @[simp] theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 1 0) = ite p 1 0 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_one_zero RingHom.map_ite_one_zero /-- `f : α →+* β` has a trivial codomain iff `f 1 = 0`. -/ theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 := by rw [map_one, eq_comm] #align ring_hom.codomain_trivial_iff_map_one_eq_zero RingHom.codomain_trivial_iff_map_one_eq_zero /-- `f : α →+* β` has a trivial codomain iff it has a trivial range. -/ theorem codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 := f.codomain_trivial_iff_map_one_eq_zero.trans ⟨fun h x => by rw [← mul_one x, map_mul, h, mul_zero], fun h => h 1⟩ #align ring_hom.codomain_trivial_iff_range_trivial RingHom.codomain_trivial_iff_range_trivial /-- `f : α →+* β` doesn't map `1` to `0` if `β` is nontrivial -/ theorem map_one_ne_zero [Nontrivial β] : f 1 ≠ 0 := mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one #align ring_hom.map_one_ne_zero RingHom.map_one_ne_zero /-- If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. -/ theorem domain_nontrivial [Nontrivial β] : Nontrivial α := ⟨⟨1, 0, mt (fun h => show f 1 = 0 by rw [h, map_zero]) f.map_one_ne_zero⟩⟩ #align ring_hom.domain_nontrivial RingHom.domain_nontrivial theorem codomain_trivial (f : α →+* β) [h : Subsingleton α] : Subsingleton β := (subsingleton_or_nontrivial β).resolve_right fun _ => not_nontrivial_iff_subsingleton.mpr h f.domain_nontrivial #align ring_hom.codomain_trivial RingHom.codomain_trivial end /-- Ring homomorphisms preserve additive inverse. -/ protected theorem map_neg [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x : α) : f (-x) = -f x := map_neg f x #align ring_hom.map_neg RingHom.map_neg /-- Ring homomorphisms preserve subtraction. -/ protected theorem map_sub [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x y : α) : f (x - y) = f x - f y := map_sub f x y #align ring_hom.map_sub RingHom.map_sub /-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/ def mk' [NonAssocSemiring α] [NonAssocRing β] (f : α →* β) (map_add : ∀ a b, f (a + b) = f a + f b) : α →+* β := { AddMonoidHom.mk' f map_add, f with } #align ring_hom.mk' RingHom.mk' variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} /-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;>

intros

/-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;>

Mathlib.Algebra.Ring.Hom.Defs.631_0.KyHvVYrIs9pW9ZQ

/-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α

Mathlib_Algebra_Ring_Hom_Defs

case refine'_2 F : Type u_1 α✝ : Type u_2 β : Type u_3 γ : Type u_4 x✝¹ : NonAssocSemiring α✝ x✝ : NonAssocSemiring β α : Type u_5 inst✝ : NonAssocSemiring α ⊢ ∀ (x y : α), OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } (x * y) = OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } x * OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } y

/- Copyright (c) 2019 Amelia Livingston. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Amelia Livingston, Jireh Loreaux -/ import Mathlib.Algebra.Ring.Defs import Mathlib.Algebra.Ring.Basic import Mathlib.Data.Pi.Algebra #align_import algebra.hom.ring from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9" /-! # Homomorphisms of semirings and rings This file defines bundled homomorphisms of (non-unital) semirings and rings. As with monoid and groups, we use the same structure `RingHom a β`, a.k.a. `α →+* β`, for both types of homomorphisms. ## Main definitions * `NonUnitalRingHom`: Non-unital (semi)ring homomorphisms. Additive monoid homomorphism which preserve multiplication. * `RingHom`: (Semi)ring homomorphisms. Monoid homomorphisms which are also additive monoid homomorphism. ## Notations * `→ₙ+*`: Non-unital (semi)ring homs * `→+*`: (Semi)ring homs ## Implementation notes * There's a coercion from bundled homs to fun, and the canonical notation is to use the bundled hom as a function via this coercion. * There is no `SemiringHom` -- the idea is that `RingHom` is used. The constructor for a `RingHom` between semirings needs a proof of `map_zero`, `map_one` and `map_add` as well as `map_mul`; a separate constructor `RingHom.mk'` will construct ring homs between rings from monoid homs given only a proof that addition is preserved. ## Tags `RingHom`, `SemiringHom` -/ open Function variable {F α β γ : Type*} /-- Bundled non-unital semiring homomorphisms `α →ₙ+* β`; use this for bundled non-unital ring homomorphisms too. When possible, instead of parametrizing results over `(f : α →ₙ+* β)`, you should parametrize over `(F : Type*) [NonUnitalRingHomClass F α β] (f : F)`. When you extend this structure, make sure to extend `NonUnitalRingHomClass`. -/ structure NonUnitalRingHom (α β : Type*) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends α →ₙ* β, α →+ β #align non_unital_ring_hom NonUnitalRingHom /-- `α →ₙ+* β` denotes the type of non-unital ring homomorphisms from `α` to `β`. -/ infixr:25 " →ₙ+* " => NonUnitalRingHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as a semigroup homomorphism `α →ₙ* β`. The `simp`-normal form is `(f : α →ₙ* β)`. -/ add_decl_doc NonUnitalRingHom.toMulHom #align non_unital_ring_hom.to_mul_hom NonUnitalRingHom.toMulHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc NonUnitalRingHom.toAddMonoidHom #align non_unital_ring_hom.to_add_monoid_hom NonUnitalRingHom.toAddMonoidHom section NonUnitalRingHomClass /-- `NonUnitalRingHomClass F α β` states that `F` is a type of non-unital (semi)ring homomorphisms. You should extend this class when you extend `NonUnitalRingHom`. -/ class NonUnitalRingHomClass (F : Type*) (α β : outParam (Type*)) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends MulHomClass F α β, AddMonoidHomClass F α β #align non_unital_ring_hom_class NonUnitalRingHomClass variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] [NonUnitalRingHomClass F α β] /-- Turn an element of a type `F` satisfying `NonUnitalRingHomClass F α β` into an actual `NonUnitalRingHom`. This is declared as the default coercion from `F` to `α →ₙ+* β`. -/ @[coe] def NonUnitalRingHomClass.toNonUnitalRingHom (f : F) : α →ₙ+* β := { (f : α →ₙ* β), (f : α →+ β) with } /-- Any type satisfying `NonUnitalRingHomClass` can be cast into `NonUnitalRingHom` via `NonUnitalRingHomClass.toNonUnitalRingHom`. -/ instance : CoeTC F (α →ₙ+* β) := ⟨NonUnitalRingHomClass.toNonUnitalRingHom⟩ end NonUnitalRingHomClass namespace NonUnitalRingHom section coe variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] instance : NonUnitalRingHomClass (α →ₙ+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := NonUnitalRingHom.map_add' map_zero := NonUnitalRingHom.map_zero' map_mul f := f.map_mul' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -/ -- instance : CoeFun (α →ₙ+* β) fun _ => α → β := -- ⟨fun f => f.toFun⟩ -- Porting note: removed due to new `coe` in Lean4 #noalign non_unital_ring_hom.to_fun_eq_coe #noalign non_unital_ring_hom.coe_mk #noalign non_unital_ring_hom.coe_coe initialize_simps_projections NonUnitalRingHom (toFun → apply) @[simp] theorem coe_toMulHom (f : α →ₙ+* β) : ⇑f.toMulHom = f := rfl #align non_unital_ring_hom.coe_to_mul_hom NonUnitalRingHom.coe_toMulHom @[simp] theorem coe_mulHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →ₙ* β) = ⟨f, h₁⟩ := rfl #align non_unital_ring_hom.coe_mul_hom_mk NonUnitalRingHom.coe_mulHom_mk theorem coe_toAddMonoidHom (f : α →ₙ+* β) : ⇑f.toAddMonoidHom = f := rfl #align non_unital_ring_hom.coe_to_add_monoid_hom NonUnitalRingHom.coe_toAddMonoidHom @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →+ β) = ⟨⟨f, h₂⟩, h₃⟩ := rfl #align non_unital_ring_hom.coe_add_monoid_hom_mk NonUnitalRingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : α →ₙ+* β := { f.toMulHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align non_unital_ring_hom.copy NonUnitalRingHom.copy @[simp] theorem coe_copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align non_unital_ring_hom.coe_copy NonUnitalRingHom.coe_copy theorem copy_eq (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align non_unital_ring_hom.copy_eq NonUnitalRingHom.copy_eq end coe section variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] variable (f : α →ₙ+* β) {x y : α} @[ext] theorem ext ⦃f g : α →ₙ+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align non_unital_ring_hom.ext NonUnitalRingHom.ext theorem ext_iff {f g : α →ₙ+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align non_unital_ring_hom.ext_iff NonUnitalRingHom.ext_iff @[simp] theorem mk_coe (f : α →ₙ+* β) (h₁ h₂ h₃) : NonUnitalRingHom.mk (MulHom.mk f h₁) h₂ h₃ = f := ext fun _ => rfl #align non_unital_ring_hom.mk_coe NonUnitalRingHom.mk_coe theorem coe_addMonoidHom_injective : Injective fun f : α →ₙ+* β => (f : α →+ β) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align non_unital_ring_hom.coe_add_monoid_hom_injective NonUnitalRingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_mulHom_injective : Injective fun f : α →ₙ+* β => (f : α →ₙ* β) := fun _ _ h => ext <| MulHom.congr_fun h #align non_unital_ring_hom.coe_mul_hom_injective NonUnitalRingHom.coe_mulHom_injective end variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/ protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by refine' { toFun := id.. } <;> intros <;> rfl #align non_unital_ring_hom.id NonUnitalRingHom.id instance : Zero (α →ₙ+* β) := ⟨{ toFun := 0, map_mul' := fun _ _ => (mul_zero (0 : β)).symm, map_zero' := rfl, map_add' := fun _ _ => (add_zero (0 : β)).symm }⟩ instance : Inhabited (α →ₙ+* β) := ⟨0⟩ @[simp] theorem coe_zero : ⇑(0 : α →ₙ+* β) = 0 := rfl #align non_unital_ring_hom.coe_zero NonUnitalRingHom.coe_zero @[simp] theorem zero_apply (x : α) : (0 : α →ₙ+* β) x = 0 := rfl #align non_unital_ring_hom.zero_apply NonUnitalRingHom.zero_apply @[simp] theorem id_apply (x : α) : NonUnitalRingHom.id α x = x := rfl #align non_unital_ring_hom.id_apply NonUnitalRingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (NonUnitalRingHom.id α : α →+ α) = AddMonoidHom.id α := rfl #align non_unital_ring_hom.coe_add_monoid_hom_id NonUnitalRingHom.coe_addMonoidHom_id @[simp] theorem coe_mulHom_id : (NonUnitalRingHom.id α : α →ₙ* α) = MulHom.id α := rfl #align non_unital_ring_hom.coe_mul_hom_id NonUnitalRingHom.coe_mulHom_id variable [NonUnitalNonAssocSemiring γ] /-- Composition of non-unital ring homomorphisms is a non-unital ring homomorphism. -/ def comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : α →ₙ+* γ := { g.toMulHom.comp f.toMulHom, g.toAddMonoidHom.comp f.toAddMonoidHom with } #align non_unital_ring_hom.comp NonUnitalRingHom.comp /-- Composition of non-unital ring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonUnitalNonAssocSemiring δ} (f : α →ₙ+* β) (g : β →ₙ+* γ) (h : γ →ₙ+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align non_unital_ring_hom.comp_assoc NonUnitalRingHom.comp_assoc @[simp] theorem coe_comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : ⇑(g.comp f) = g ∘ f := rfl #align non_unital_ring_hom.coe_comp NonUnitalRingHom.coe_comp @[simp] theorem comp_apply (g : β →ₙ+* γ) (f : α →ₙ+* β) (x : α) : g.comp f x = g (f x) := rfl #align non_unital_ring_hom.comp_apply NonUnitalRingHom.comp_apply variable (g : β →ₙ+* γ) (f : α →ₙ+* β) @[simp] theorem coe_comp_addMonoidHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : AddMonoidHom.mk ⟨g ∘ f, (g.comp f).map_zero'⟩ (g.comp f).map_add' = (g : β →+ γ).comp f := rfl #align non_unital_ring_hom.coe_comp_add_monoid_hom NonUnitalRingHom.coe_comp_addMonoidHom @[simp] theorem coe_comp_mulHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : MulHom.mk (g ∘ f) (g.comp f).map_mul' = (g : β →ₙ* γ).comp f := rfl #align non_unital_ring_hom.coe_comp_mul_hom NonUnitalRingHom.coe_comp_mulHom @[simp] theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 := by ext simp #align non_unital_ring_hom.comp_zero NonUnitalRingHom.comp_zero @[simp] theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 := by ext rfl #align non_unital_ring_hom.zero_comp NonUnitalRingHom.zero_comp @[simp] theorem comp_id (f : α →ₙ+* β) : f.comp (NonUnitalRingHom.id α) = f := ext fun _ => rfl #align non_unital_ring_hom.comp_id NonUnitalRingHom.comp_id @[simp] theorem id_comp (f : α →ₙ+* β) : (NonUnitalRingHom.id β).comp f = f := ext fun _ => rfl #align non_unital_ring_hom.id_comp NonUnitalRingHom.id_comp instance : MonoidWithZero (α →ₙ+* α) where one := NonUnitalRingHom.id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ zero := 0 mul_zero := comp_zero zero_mul := zero_comp theorem one_def : (1 : α →ₙ+* α) = NonUnitalRingHom.id α := rfl #align non_unital_ring_hom.one_def NonUnitalRingHom.one_def @[simp] theorem coe_one : ⇑(1 : α →ₙ+* α) = id := rfl #align non_unital_ring_hom.coe_one NonUnitalRingHom.coe_one theorem mul_def (f g : α →ₙ+* α) : f * g = f.comp g := rfl #align non_unital_ring_hom.mul_def NonUnitalRingHom.mul_def @[simp] theorem coe_mul (f g : α →ₙ+* α) : ⇑(f * g) = f ∘ g := rfl #align non_unital_ring_hom.coe_mul NonUnitalRingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →ₙ+* γ} {f : α →ₙ+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_right NonUnitalRingHom.cancel_right @[simp] theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_left NonUnitalRingHom.cancel_left end NonUnitalRingHom /-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too. This extends from both `MonoidHom` and `MonoidWithZeroHom` in order to put the fields in a sensible order, even though `MonoidWithZeroHom` already extends `MonoidHom`. -/ structure RingHom (α : Type*) (β : Type*) [NonAssocSemiring α] [NonAssocSemiring β] extends α →* β, α →+ β, α →ₙ+* β, α →*₀ β #align ring_hom RingHom /-- `α →+* β` denotes the type of ring homomorphisms from `α` to `β`. -/ infixr:25 " →+* " => RingHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid with zero homomorphism `α →*₀ β`. The `simp`-normal form is `(f : α →*₀ β)`. -/ add_decl_doc RingHom.toMonoidWithZeroHom #align ring_hom.to_monoid_with_zero_hom RingHom.toMonoidWithZeroHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid homomorphism `α →* β`. The `simp`-normal form is `(f : α →* β)`. -/ add_decl_doc RingHom.toMonoidHom #align ring_hom.to_monoid_hom RingHom.toMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc RingHom.toAddMonoidHom #align ring_hom.to_add_monoid_hom RingHom.toAddMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a non-unital ring homomorphism `α →ₙ+* β`. The `simp`-normal form is `(f : α →ₙ+* β)`. -/ add_decl_doc RingHom.toNonUnitalRingHom #align ring_hom.to_non_unital_ring_hom RingHom.toNonUnitalRingHom section RingHomClass /-- `RingHomClass F α β` states that `F` is a type of (semi)ring homomorphisms. You should extend this class when you extend `RingHom`. This extends from both `MonoidHomClass` and `MonoidWithZeroHomClass` in order to put the fields in a sensible order, even though `MonoidWithZeroHomClass` already extends `MonoidHomClass`. -/ class RingHomClass (F : Type*) (α β : outParam (Type*)) [NonAssocSemiring α] [NonAssocSemiring β] extends MonoidHomClass F α β, AddMonoidHomClass F α β, MonoidWithZeroHomClass F α β #align ring_hom_class RingHomClass set_option linter.deprecated false in /-- Ring homomorphisms preserve `bit1`. -/ @[simp] lemma map_bit1 [NonAssocSemiring α] [NonAssocSemiring β] [RingHomClass F α β] (f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) := by simp [bit1] #align map_bit1 map_bit1 -- Porting note: marked `{}` rather than `[]` to prevent dangerous instances variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} [RingHomClass F α β] /-- Turn an element of a type `F` satisfying `RingHomClass F α β` into an actual `RingHom`. This is declared as the default coercion from `F` to `α →+* β`. -/ @[coe] def RingHomClass.toRingHom (f : F) : α →+* β := { (f : α →* β), (f : α →+ β) with } /-- Any type satisfying `RingHomClass` can be cast into `RingHom` via `RingHomClass.toRingHom`. -/ instance : CoeTC F (α →+* β) := ⟨RingHomClass.toRingHom⟩ instance (priority := 100) RingHomClass.toNonUnitalRingHomClass : NonUnitalRingHomClass F α β := { ‹RingHomClass F α β› with } #align ring_hom_class.to_non_unital_ring_hom_class RingHomClass.toNonUnitalRingHomClass end RingHomClass namespace RingHom section coe /-! Throughout this section, some `Semiring` arguments are specified with `{}` instead of `[]`. See note [implicit instance arguments]. -/ variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} instance instRingHomClass : RingHomClass (α →+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := RingHom.map_add' map_zero := RingHom.map_zero' map_mul f := f.map_mul' map_one f := f.map_one' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -- -/ -- instance : CoeFun (α →+* β) fun _ => α → β := -- ⟨RingHom.toFun⟩ initialize_simps_projections RingHom (toFun → apply) -- Porting note: is this lemma still needed in Lean4? -- Porting note: because `f.toFun` really means `f.toMonoidHom.toOneHom.toFun` and -- `toMonoidHom_eq_coe` wants to simplify `f.toMonoidHom` to `(↑f : M →* N)`, this can't -- be a simp lemma anymore -- @[simp] theorem toFun_eq_coe (f : α →+* β) : f.toFun = f := rfl #align ring_hom.to_fun_eq_coe RingHom.toFun_eq_coe @[simp] theorem coe_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α → β) = f := rfl #align ring_hom.coe_mk RingHom.coe_mk @[simp] theorem coe_coe {F : Type*} [RingHomClass F α β] (f : F) : ((f : α →+* β) : α → β) = f := rfl #align ring_hom.coe_coe RingHom.coe_coe attribute [coe] RingHom.toMonoidHom instance coeToMonoidHom : Coe (α →+* β) (α →* β) := ⟨RingHom.toMonoidHom⟩ #align ring_hom.has_coe_monoid_hom RingHom.coeToMonoidHom -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_monoid_hom @[simp] theorem toMonoidHom_eq_coe (f : α →+* β) : f.toMonoidHom = f := rfl #align ring_hom.to_monoid_hom_eq_coe RingHom.toMonoidHom_eq_coe -- Porting note: this can't be a simp lemma anymore -- @[simp] theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f := by rfl #align ring_hom.to_monoid_with_zero_hom_eq_coe RingHom.toMonoidWithZeroHom_eq_coe @[simp] theorem coe_monoidHom_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α →* β) = f := rfl #align ring_hom.coe_monoid_hom_mk RingHom.coe_monoidHom_mk -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_add_monoid_hom @[simp] theorem toAddMonoidHom_eq_coe (f : α →+* β) : f.toAddMonoidHom = f := rfl #align ring_hom.to_add_monoid_hom_eq_coe RingHom.toAddMonoidHom_eq_coe @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃ h₄) : ((⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨⟨f, h₃⟩, h₄⟩ := rfl #align ring_hom.coe_add_monoid_hom_mk RingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ def copy (f : α →+* β) (f' : α → β) (h : f' = f) : α →+* β := { f.toMonoidWithZeroHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align ring_hom.copy RingHom.copy @[simp] theorem coe_copy (f : α →+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align ring_hom.coe_copy RingHom.coe_copy theorem copy_eq (f : α →+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align ring_hom.copy_eq RingHom.copy_eq end coe section variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} (f : α →+* β) {x y : α} theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x := FunLike.congr_fun h x #align ring_hom.congr_fun RingHom.congr_fun theorem congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y := FunLike.congr_arg f h #align ring_hom.congr_arg RingHom.congr_arg theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g := FunLike.coe_injective h #align ring_hom.coe_inj RingHom.coe_inj @[ext] theorem ext ⦃f g : α →+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align ring_hom.ext RingHom.ext theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align ring_hom.ext_iff RingHom.ext_iff @[simp] theorem mk_coe (f : α →+* β) (h₁ h₂ h₃ h₄) : RingHom.mk ⟨⟨f, h₁⟩, h₂⟩ h₃ h₄ = f := ext fun _ => rfl #align ring_hom.mk_coe RingHom.mk_coe theorem coe_addMonoidHom_injective : Injective (fun f : α →+* β => (f : α →+ β)) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align ring_hom.coe_add_monoid_hom_injective RingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_monoidHom_injective : Injective (fun f : α →+* β => (f : α →* β)) := fun _ _ h => ext <| MonoidHom.congr_fun h #align ring_hom.coe_monoid_hom_injective RingHom.coe_monoidHom_injective /-- Ring homomorphisms map zero to zero. -/ protected theorem map_zero (f : α →+* β) : f 0 = 0 := map_zero f #align ring_hom.map_zero RingHom.map_zero /-- Ring homomorphisms map one to one. -/ protected theorem map_one (f : α →+* β) : f 1 = 1 := map_one f #align ring_hom.map_one RingHom.map_one /-- Ring homomorphisms preserve addition. -/ protected theorem map_add (f : α →+* β) : ∀ a b, f (a + b) = f a + f b := map_add f #align ring_hom.map_add RingHom.map_add /-- Ring homomorphisms preserve multiplication. -/ protected theorem map_mul (f : α →+* β) : ∀ a b, f (a * b) = f a * f b := map_mul f #align ring_hom.map_mul RingHom.map_mul @[simp] theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 0 1) = ite p 0 1 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_zero_one RingHom.map_ite_zero_one @[simp] theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 1 0) = ite p 1 0 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_one_zero RingHom.map_ite_one_zero /-- `f : α →+* β` has a trivial codomain iff `f 1 = 0`. -/ theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 := by rw [map_one, eq_comm] #align ring_hom.codomain_trivial_iff_map_one_eq_zero RingHom.codomain_trivial_iff_map_one_eq_zero /-- `f : α →+* β` has a trivial codomain iff it has a trivial range. -/ theorem codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 := f.codomain_trivial_iff_map_one_eq_zero.trans ⟨fun h x => by rw [← mul_one x, map_mul, h, mul_zero], fun h => h 1⟩ #align ring_hom.codomain_trivial_iff_range_trivial RingHom.codomain_trivial_iff_range_trivial /-- `f : α →+* β` doesn't map `1` to `0` if `β` is nontrivial -/ theorem map_one_ne_zero [Nontrivial β] : f 1 ≠ 0 := mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one #align ring_hom.map_one_ne_zero RingHom.map_one_ne_zero /-- If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. -/ theorem domain_nontrivial [Nontrivial β] : Nontrivial α := ⟨⟨1, 0, mt (fun h => show f 1 = 0 by rw [h, map_zero]) f.map_one_ne_zero⟩⟩ #align ring_hom.domain_nontrivial RingHom.domain_nontrivial theorem codomain_trivial (f : α →+* β) [h : Subsingleton α] : Subsingleton β := (subsingleton_or_nontrivial β).resolve_right fun _ => not_nontrivial_iff_subsingleton.mpr h f.domain_nontrivial #align ring_hom.codomain_trivial RingHom.codomain_trivial end /-- Ring homomorphisms preserve additive inverse. -/ protected theorem map_neg [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x : α) : f (-x) = -f x := map_neg f x #align ring_hom.map_neg RingHom.map_neg /-- Ring homomorphisms preserve subtraction. -/ protected theorem map_sub [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x y : α) : f (x - y) = f x - f y := map_sub f x y #align ring_hom.map_sub RingHom.map_sub /-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/ def mk' [NonAssocSemiring α] [NonAssocRing β] (f : α →* β) (map_add : ∀ a b, f (a + b) = f a + f b) : α →+* β := { AddMonoidHom.mk' f map_add, f with } #align ring_hom.mk' RingHom.mk' variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} /-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;>

intros

/-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;>

Mathlib.Algebra.Ring.Hom.Defs.631_0.KyHvVYrIs9pW9ZQ

/-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α

Mathlib_Algebra_Ring_Hom_Defs

case refine'_3 F : Type u_1 α✝ : Type u_2 β : Type u_3 γ : Type u_4 x✝¹ : NonAssocSemiring α✝ x✝ : NonAssocSemiring β α : Type u_5 inst✝ : NonAssocSemiring α ⊢ OneHom.toFun (↑{ toOneHom := { toFun := _root_.id, map_one' := ?refine'_1 }, map_mul' := (_ : ∀ (x y : α), OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } (x * y) = OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } x * OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } y) }) 0 = 0

/- Copyright (c) 2019 Amelia Livingston. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Amelia Livingston, Jireh Loreaux -/ import Mathlib.Algebra.Ring.Defs import Mathlib.Algebra.Ring.Basic import Mathlib.Data.Pi.Algebra #align_import algebra.hom.ring from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9" /-! # Homomorphisms of semirings and rings This file defines bundled homomorphisms of (non-unital) semirings and rings. As with monoid and groups, we use the same structure `RingHom a β`, a.k.a. `α →+* β`, for both types of homomorphisms. ## Main definitions * `NonUnitalRingHom`: Non-unital (semi)ring homomorphisms. Additive monoid homomorphism which preserve multiplication. * `RingHom`: (Semi)ring homomorphisms. Monoid homomorphisms which are also additive monoid homomorphism. ## Notations * `→ₙ+*`: Non-unital (semi)ring homs * `→+*`: (Semi)ring homs ## Implementation notes * There's a coercion from bundled homs to fun, and the canonical notation is to use the bundled hom as a function via this coercion. * There is no `SemiringHom` -- the idea is that `RingHom` is used. The constructor for a `RingHom` between semirings needs a proof of `map_zero`, `map_one` and `map_add` as well as `map_mul`; a separate constructor `RingHom.mk'` will construct ring homs between rings from monoid homs given only a proof that addition is preserved. ## Tags `RingHom`, `SemiringHom` -/ open Function variable {F α β γ : Type*} /-- Bundled non-unital semiring homomorphisms `α →ₙ+* β`; use this for bundled non-unital ring homomorphisms too. When possible, instead of parametrizing results over `(f : α →ₙ+* β)`, you should parametrize over `(F : Type*) [NonUnitalRingHomClass F α β] (f : F)`. When you extend this structure, make sure to extend `NonUnitalRingHomClass`. -/ structure NonUnitalRingHom (α β : Type*) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends α →ₙ* β, α →+ β #align non_unital_ring_hom NonUnitalRingHom /-- `α →ₙ+* β` denotes the type of non-unital ring homomorphisms from `α` to `β`. -/ infixr:25 " →ₙ+* " => NonUnitalRingHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as a semigroup homomorphism `α →ₙ* β`. The `simp`-normal form is `(f : α →ₙ* β)`. -/ add_decl_doc NonUnitalRingHom.toMulHom #align non_unital_ring_hom.to_mul_hom NonUnitalRingHom.toMulHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc NonUnitalRingHom.toAddMonoidHom #align non_unital_ring_hom.to_add_monoid_hom NonUnitalRingHom.toAddMonoidHom section NonUnitalRingHomClass /-- `NonUnitalRingHomClass F α β` states that `F` is a type of non-unital (semi)ring homomorphisms. You should extend this class when you extend `NonUnitalRingHom`. -/ class NonUnitalRingHomClass (F : Type*) (α β : outParam (Type*)) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends MulHomClass F α β, AddMonoidHomClass F α β #align non_unital_ring_hom_class NonUnitalRingHomClass variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] [NonUnitalRingHomClass F α β] /-- Turn an element of a type `F` satisfying `NonUnitalRingHomClass F α β` into an actual `NonUnitalRingHom`. This is declared as the default coercion from `F` to `α →ₙ+* β`. -/ @[coe] def NonUnitalRingHomClass.toNonUnitalRingHom (f : F) : α →ₙ+* β := { (f : α →ₙ* β), (f : α →+ β) with } /-- Any type satisfying `NonUnitalRingHomClass` can be cast into `NonUnitalRingHom` via `NonUnitalRingHomClass.toNonUnitalRingHom`. -/ instance : CoeTC F (α →ₙ+* β) := ⟨NonUnitalRingHomClass.toNonUnitalRingHom⟩ end NonUnitalRingHomClass namespace NonUnitalRingHom section coe variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] instance : NonUnitalRingHomClass (α →ₙ+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := NonUnitalRingHom.map_add' map_zero := NonUnitalRingHom.map_zero' map_mul f := f.map_mul' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -/ -- instance : CoeFun (α →ₙ+* β) fun _ => α → β := -- ⟨fun f => f.toFun⟩ -- Porting note: removed due to new `coe` in Lean4 #noalign non_unital_ring_hom.to_fun_eq_coe #noalign non_unital_ring_hom.coe_mk #noalign non_unital_ring_hom.coe_coe initialize_simps_projections NonUnitalRingHom (toFun → apply) @[simp] theorem coe_toMulHom (f : α →ₙ+* β) : ⇑f.toMulHom = f := rfl #align non_unital_ring_hom.coe_to_mul_hom NonUnitalRingHom.coe_toMulHom @[simp] theorem coe_mulHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →ₙ* β) = ⟨f, h₁⟩ := rfl #align non_unital_ring_hom.coe_mul_hom_mk NonUnitalRingHom.coe_mulHom_mk theorem coe_toAddMonoidHom (f : α →ₙ+* β) : ⇑f.toAddMonoidHom = f := rfl #align non_unital_ring_hom.coe_to_add_monoid_hom NonUnitalRingHom.coe_toAddMonoidHom @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →+ β) = ⟨⟨f, h₂⟩, h₃⟩ := rfl #align non_unital_ring_hom.coe_add_monoid_hom_mk NonUnitalRingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : α →ₙ+* β := { f.toMulHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align non_unital_ring_hom.copy NonUnitalRingHom.copy @[simp] theorem coe_copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align non_unital_ring_hom.coe_copy NonUnitalRingHom.coe_copy theorem copy_eq (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align non_unital_ring_hom.copy_eq NonUnitalRingHom.copy_eq end coe section variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] variable (f : α →ₙ+* β) {x y : α} @[ext] theorem ext ⦃f g : α →ₙ+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align non_unital_ring_hom.ext NonUnitalRingHom.ext theorem ext_iff {f g : α →ₙ+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align non_unital_ring_hom.ext_iff NonUnitalRingHom.ext_iff @[simp] theorem mk_coe (f : α →ₙ+* β) (h₁ h₂ h₃) : NonUnitalRingHom.mk (MulHom.mk f h₁) h₂ h₃ = f := ext fun _ => rfl #align non_unital_ring_hom.mk_coe NonUnitalRingHom.mk_coe theorem coe_addMonoidHom_injective : Injective fun f : α →ₙ+* β => (f : α →+ β) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align non_unital_ring_hom.coe_add_monoid_hom_injective NonUnitalRingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_mulHom_injective : Injective fun f : α →ₙ+* β => (f : α →ₙ* β) := fun _ _ h => ext <| MulHom.congr_fun h #align non_unital_ring_hom.coe_mul_hom_injective NonUnitalRingHom.coe_mulHom_injective end variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/ protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by refine' { toFun := id.. } <;> intros <;> rfl #align non_unital_ring_hom.id NonUnitalRingHom.id instance : Zero (α →ₙ+* β) := ⟨{ toFun := 0, map_mul' := fun _ _ => (mul_zero (0 : β)).symm, map_zero' := rfl, map_add' := fun _ _ => (add_zero (0 : β)).symm }⟩ instance : Inhabited (α →ₙ+* β) := ⟨0⟩ @[simp] theorem coe_zero : ⇑(0 : α →ₙ+* β) = 0 := rfl #align non_unital_ring_hom.coe_zero NonUnitalRingHom.coe_zero @[simp] theorem zero_apply (x : α) : (0 : α →ₙ+* β) x = 0 := rfl #align non_unital_ring_hom.zero_apply NonUnitalRingHom.zero_apply @[simp] theorem id_apply (x : α) : NonUnitalRingHom.id α x = x := rfl #align non_unital_ring_hom.id_apply NonUnitalRingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (NonUnitalRingHom.id α : α →+ α) = AddMonoidHom.id α := rfl #align non_unital_ring_hom.coe_add_monoid_hom_id NonUnitalRingHom.coe_addMonoidHom_id @[simp] theorem coe_mulHom_id : (NonUnitalRingHom.id α : α →ₙ* α) = MulHom.id α := rfl #align non_unital_ring_hom.coe_mul_hom_id NonUnitalRingHom.coe_mulHom_id variable [NonUnitalNonAssocSemiring γ] /-- Composition of non-unital ring homomorphisms is a non-unital ring homomorphism. -/ def comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : α →ₙ+* γ := { g.toMulHom.comp f.toMulHom, g.toAddMonoidHom.comp f.toAddMonoidHom with } #align non_unital_ring_hom.comp NonUnitalRingHom.comp /-- Composition of non-unital ring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonUnitalNonAssocSemiring δ} (f : α →ₙ+* β) (g : β →ₙ+* γ) (h : γ →ₙ+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align non_unital_ring_hom.comp_assoc NonUnitalRingHom.comp_assoc @[simp] theorem coe_comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : ⇑(g.comp f) = g ∘ f := rfl #align non_unital_ring_hom.coe_comp NonUnitalRingHom.coe_comp @[simp] theorem comp_apply (g : β →ₙ+* γ) (f : α →ₙ+* β) (x : α) : g.comp f x = g (f x) := rfl #align non_unital_ring_hom.comp_apply NonUnitalRingHom.comp_apply variable (g : β →ₙ+* γ) (f : α →ₙ+* β) @[simp] theorem coe_comp_addMonoidHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : AddMonoidHom.mk ⟨g ∘ f, (g.comp f).map_zero'⟩ (g.comp f).map_add' = (g : β →+ γ).comp f := rfl #align non_unital_ring_hom.coe_comp_add_monoid_hom NonUnitalRingHom.coe_comp_addMonoidHom @[simp] theorem coe_comp_mulHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : MulHom.mk (g ∘ f) (g.comp f).map_mul' = (g : β →ₙ* γ).comp f := rfl #align non_unital_ring_hom.coe_comp_mul_hom NonUnitalRingHom.coe_comp_mulHom @[simp] theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 := by ext simp #align non_unital_ring_hom.comp_zero NonUnitalRingHom.comp_zero @[simp] theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 := by ext rfl #align non_unital_ring_hom.zero_comp NonUnitalRingHom.zero_comp @[simp] theorem comp_id (f : α →ₙ+* β) : f.comp (NonUnitalRingHom.id α) = f := ext fun _ => rfl #align non_unital_ring_hom.comp_id NonUnitalRingHom.comp_id @[simp] theorem id_comp (f : α →ₙ+* β) : (NonUnitalRingHom.id β).comp f = f := ext fun _ => rfl #align non_unital_ring_hom.id_comp NonUnitalRingHom.id_comp instance : MonoidWithZero (α →ₙ+* α) where one := NonUnitalRingHom.id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ zero := 0 mul_zero := comp_zero zero_mul := zero_comp theorem one_def : (1 : α →ₙ+* α) = NonUnitalRingHom.id α := rfl #align non_unital_ring_hom.one_def NonUnitalRingHom.one_def @[simp] theorem coe_one : ⇑(1 : α →ₙ+* α) = id := rfl #align non_unital_ring_hom.coe_one NonUnitalRingHom.coe_one theorem mul_def (f g : α →ₙ+* α) : f * g = f.comp g := rfl #align non_unital_ring_hom.mul_def NonUnitalRingHom.mul_def @[simp] theorem coe_mul (f g : α →ₙ+* α) : ⇑(f * g) = f ∘ g := rfl #align non_unital_ring_hom.coe_mul NonUnitalRingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →ₙ+* γ} {f : α →ₙ+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_right NonUnitalRingHom.cancel_right @[simp] theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_left NonUnitalRingHom.cancel_left end NonUnitalRingHom /-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too. This extends from both `MonoidHom` and `MonoidWithZeroHom` in order to put the fields in a sensible order, even though `MonoidWithZeroHom` already extends `MonoidHom`. -/ structure RingHom (α : Type*) (β : Type*) [NonAssocSemiring α] [NonAssocSemiring β] extends α →* β, α →+ β, α →ₙ+* β, α →*₀ β #align ring_hom RingHom /-- `α →+* β` denotes the type of ring homomorphisms from `α` to `β`. -/ infixr:25 " →+* " => RingHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid with zero homomorphism `α →*₀ β`. The `simp`-normal form is `(f : α →*₀ β)`. -/ add_decl_doc RingHom.toMonoidWithZeroHom #align ring_hom.to_monoid_with_zero_hom RingHom.toMonoidWithZeroHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid homomorphism `α →* β`. The `simp`-normal form is `(f : α →* β)`. -/ add_decl_doc RingHom.toMonoidHom #align ring_hom.to_monoid_hom RingHom.toMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc RingHom.toAddMonoidHom #align ring_hom.to_add_monoid_hom RingHom.toAddMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a non-unital ring homomorphism `α →ₙ+* β`. The `simp`-normal form is `(f : α →ₙ+* β)`. -/ add_decl_doc RingHom.toNonUnitalRingHom #align ring_hom.to_non_unital_ring_hom RingHom.toNonUnitalRingHom section RingHomClass /-- `RingHomClass F α β` states that `F` is a type of (semi)ring homomorphisms. You should extend this class when you extend `RingHom`. This extends from both `MonoidHomClass` and `MonoidWithZeroHomClass` in order to put the fields in a sensible order, even though `MonoidWithZeroHomClass` already extends `MonoidHomClass`. -/ class RingHomClass (F : Type*) (α β : outParam (Type*)) [NonAssocSemiring α] [NonAssocSemiring β] extends MonoidHomClass F α β, AddMonoidHomClass F α β, MonoidWithZeroHomClass F α β #align ring_hom_class RingHomClass set_option linter.deprecated false in /-- Ring homomorphisms preserve `bit1`. -/ @[simp] lemma map_bit1 [NonAssocSemiring α] [NonAssocSemiring β] [RingHomClass F α β] (f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) := by simp [bit1] #align map_bit1 map_bit1 -- Porting note: marked `{}` rather than `[]` to prevent dangerous instances variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} [RingHomClass F α β] /-- Turn an element of a type `F` satisfying `RingHomClass F α β` into an actual `RingHom`. This is declared as the default coercion from `F` to `α →+* β`. -/ @[coe] def RingHomClass.toRingHom (f : F) : α →+* β := { (f : α →* β), (f : α →+ β) with } /-- Any type satisfying `RingHomClass` can be cast into `RingHom` via `RingHomClass.toRingHom`. -/ instance : CoeTC F (α →+* β) := ⟨RingHomClass.toRingHom⟩ instance (priority := 100) RingHomClass.toNonUnitalRingHomClass : NonUnitalRingHomClass F α β := { ‹RingHomClass F α β› with } #align ring_hom_class.to_non_unital_ring_hom_class RingHomClass.toNonUnitalRingHomClass end RingHomClass namespace RingHom section coe /-! Throughout this section, some `Semiring` arguments are specified with `{}` instead of `[]`. See note [implicit instance arguments]. -/ variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} instance instRingHomClass : RingHomClass (α →+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := RingHom.map_add' map_zero := RingHom.map_zero' map_mul f := f.map_mul' map_one f := f.map_one' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -- -/ -- instance : CoeFun (α →+* β) fun _ => α → β := -- ⟨RingHom.toFun⟩ initialize_simps_projections RingHom (toFun → apply) -- Porting note: is this lemma still needed in Lean4? -- Porting note: because `f.toFun` really means `f.toMonoidHom.toOneHom.toFun` and -- `toMonoidHom_eq_coe` wants to simplify `f.toMonoidHom` to `(↑f : M →* N)`, this can't -- be a simp lemma anymore -- @[simp] theorem toFun_eq_coe (f : α →+* β) : f.toFun = f := rfl #align ring_hom.to_fun_eq_coe RingHom.toFun_eq_coe @[simp] theorem coe_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α → β) = f := rfl #align ring_hom.coe_mk RingHom.coe_mk @[simp] theorem coe_coe {F : Type*} [RingHomClass F α β] (f : F) : ((f : α →+* β) : α → β) = f := rfl #align ring_hom.coe_coe RingHom.coe_coe attribute [coe] RingHom.toMonoidHom instance coeToMonoidHom : Coe (α →+* β) (α →* β) := ⟨RingHom.toMonoidHom⟩ #align ring_hom.has_coe_monoid_hom RingHom.coeToMonoidHom -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_monoid_hom @[simp] theorem toMonoidHom_eq_coe (f : α →+* β) : f.toMonoidHom = f := rfl #align ring_hom.to_monoid_hom_eq_coe RingHom.toMonoidHom_eq_coe -- Porting note: this can't be a simp lemma anymore -- @[simp] theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f := by rfl #align ring_hom.to_monoid_with_zero_hom_eq_coe RingHom.toMonoidWithZeroHom_eq_coe @[simp] theorem coe_monoidHom_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α →* β) = f := rfl #align ring_hom.coe_monoid_hom_mk RingHom.coe_monoidHom_mk -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_add_monoid_hom @[simp] theorem toAddMonoidHom_eq_coe (f : α →+* β) : f.toAddMonoidHom = f := rfl #align ring_hom.to_add_monoid_hom_eq_coe RingHom.toAddMonoidHom_eq_coe @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃ h₄) : ((⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨⟨f, h₃⟩, h₄⟩ := rfl #align ring_hom.coe_add_monoid_hom_mk RingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ def copy (f : α →+* β) (f' : α → β) (h : f' = f) : α →+* β := { f.toMonoidWithZeroHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align ring_hom.copy RingHom.copy @[simp] theorem coe_copy (f : α →+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align ring_hom.coe_copy RingHom.coe_copy theorem copy_eq (f : α →+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align ring_hom.copy_eq RingHom.copy_eq end coe section variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} (f : α →+* β) {x y : α} theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x := FunLike.congr_fun h x #align ring_hom.congr_fun RingHom.congr_fun theorem congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y := FunLike.congr_arg f h #align ring_hom.congr_arg RingHom.congr_arg theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g := FunLike.coe_injective h #align ring_hom.coe_inj RingHom.coe_inj @[ext] theorem ext ⦃f g : α →+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align ring_hom.ext RingHom.ext theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align ring_hom.ext_iff RingHom.ext_iff @[simp] theorem mk_coe (f : α →+* β) (h₁ h₂ h₃ h₄) : RingHom.mk ⟨⟨f, h₁⟩, h₂⟩ h₃ h₄ = f := ext fun _ => rfl #align ring_hom.mk_coe RingHom.mk_coe theorem coe_addMonoidHom_injective : Injective (fun f : α →+* β => (f : α →+ β)) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align ring_hom.coe_add_monoid_hom_injective RingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_monoidHom_injective : Injective (fun f : α →+* β => (f : α →* β)) := fun _ _ h => ext <| MonoidHom.congr_fun h #align ring_hom.coe_monoid_hom_injective RingHom.coe_monoidHom_injective /-- Ring homomorphisms map zero to zero. -/ protected theorem map_zero (f : α →+* β) : f 0 = 0 := map_zero f #align ring_hom.map_zero RingHom.map_zero /-- Ring homomorphisms map one to one. -/ protected theorem map_one (f : α →+* β) : f 1 = 1 := map_one f #align ring_hom.map_one RingHom.map_one /-- Ring homomorphisms preserve addition. -/ protected theorem map_add (f : α →+* β) : ∀ a b, f (a + b) = f a + f b := map_add f #align ring_hom.map_add RingHom.map_add /-- Ring homomorphisms preserve multiplication. -/ protected theorem map_mul (f : α →+* β) : ∀ a b, f (a * b) = f a * f b := map_mul f #align ring_hom.map_mul RingHom.map_mul @[simp] theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 0 1) = ite p 0 1 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_zero_one RingHom.map_ite_zero_one @[simp] theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 1 0) = ite p 1 0 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_one_zero RingHom.map_ite_one_zero /-- `f : α →+* β` has a trivial codomain iff `f 1 = 0`. -/ theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 := by rw [map_one, eq_comm] #align ring_hom.codomain_trivial_iff_map_one_eq_zero RingHom.codomain_trivial_iff_map_one_eq_zero /-- `f : α →+* β` has a trivial codomain iff it has a trivial range. -/ theorem codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 := f.codomain_trivial_iff_map_one_eq_zero.trans ⟨fun h x => by rw [← mul_one x, map_mul, h, mul_zero], fun h => h 1⟩ #align ring_hom.codomain_trivial_iff_range_trivial RingHom.codomain_trivial_iff_range_trivial /-- `f : α →+* β` doesn't map `1` to `0` if `β` is nontrivial -/ theorem map_one_ne_zero [Nontrivial β] : f 1 ≠ 0 := mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one #align ring_hom.map_one_ne_zero RingHom.map_one_ne_zero /-- If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. -/ theorem domain_nontrivial [Nontrivial β] : Nontrivial α := ⟨⟨1, 0, mt (fun h => show f 1 = 0 by rw [h, map_zero]) f.map_one_ne_zero⟩⟩ #align ring_hom.domain_nontrivial RingHom.domain_nontrivial theorem codomain_trivial (f : α →+* β) [h : Subsingleton α] : Subsingleton β := (subsingleton_or_nontrivial β).resolve_right fun _ => not_nontrivial_iff_subsingleton.mpr h f.domain_nontrivial #align ring_hom.codomain_trivial RingHom.codomain_trivial end /-- Ring homomorphisms preserve additive inverse. -/ protected theorem map_neg [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x : α) : f (-x) = -f x := map_neg f x #align ring_hom.map_neg RingHom.map_neg /-- Ring homomorphisms preserve subtraction. -/ protected theorem map_sub [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x y : α) : f (x - y) = f x - f y := map_sub f x y #align ring_hom.map_sub RingHom.map_sub /-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/ def mk' [NonAssocSemiring α] [NonAssocRing β] (f : α →* β) (map_add : ∀ a b, f (a + b) = f a + f b) : α →+* β := { AddMonoidHom.mk' f map_add, f with } #align ring_hom.mk' RingHom.mk' variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} /-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;>

intros

/-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;>

Mathlib.Algebra.Ring.Hom.Defs.631_0.KyHvVYrIs9pW9ZQ

/-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α

Mathlib_Algebra_Ring_Hom_Defs

case refine'_4 F : Type u_1 α✝ : Type u_2 β : Type u_3 γ : Type u_4 x✝¹ : NonAssocSemiring α✝ x✝ : NonAssocSemiring β α : Type u_5 inst✝ : NonAssocSemiring α ⊢ ∀ (x y : α), OneHom.toFun (↑{ toOneHom := { toFun := _root_.id, map_one' := ?refine'_1 }, map_mul' := (_ : ∀ (x y : α), OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } (x * y) = OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } x * OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } y) }) (x + y) = OneHom.toFun (↑{ toOneHom := { toFun := _root_.id, map_one' := ?refine'_1 }, map_mul' := (_ : ∀ (x y : α), OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } (x * y) = OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } x * OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } y) }) x + OneHom.toFun (↑{ toOneHom := { toFun := _root_.id, map_one' := ?refine'_1 }, map_mul' := (_ : ∀ (x y : α), OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } (x * y) = OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } x * OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } y) }) y

/- Copyright (c) 2019 Amelia Livingston. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Amelia Livingston, Jireh Loreaux -/ import Mathlib.Algebra.Ring.Defs import Mathlib.Algebra.Ring.Basic import Mathlib.Data.Pi.Algebra #align_import algebra.hom.ring from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9" /-! # Homomorphisms of semirings and rings This file defines bundled homomorphisms of (non-unital) semirings and rings. As with monoid and groups, we use the same structure `RingHom a β`, a.k.a. `α →+* β`, for both types of homomorphisms. ## Main definitions * `NonUnitalRingHom`: Non-unital (semi)ring homomorphisms. Additive monoid homomorphism which preserve multiplication. * `RingHom`: (Semi)ring homomorphisms. Monoid homomorphisms which are also additive monoid homomorphism. ## Notations * `→ₙ+*`: Non-unital (semi)ring homs * `→+*`: (Semi)ring homs ## Implementation notes * There's a coercion from bundled homs to fun, and the canonical notation is to use the bundled hom as a function via this coercion. * There is no `SemiringHom` -- the idea is that `RingHom` is used. The constructor for a `RingHom` between semirings needs a proof of `map_zero`, `map_one` and `map_add` as well as `map_mul`; a separate constructor `RingHom.mk'` will construct ring homs between rings from monoid homs given only a proof that addition is preserved. ## Tags `RingHom`, `SemiringHom` -/ open Function variable {F α β γ : Type*} /-- Bundled non-unital semiring homomorphisms `α →ₙ+* β`; use this for bundled non-unital ring homomorphisms too. When possible, instead of parametrizing results over `(f : α →ₙ+* β)`, you should parametrize over `(F : Type*) [NonUnitalRingHomClass F α β] (f : F)`. When you extend this structure, make sure to extend `NonUnitalRingHomClass`. -/ structure NonUnitalRingHom (α β : Type*) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends α →ₙ* β, α →+ β #align non_unital_ring_hom NonUnitalRingHom /-- `α →ₙ+* β` denotes the type of non-unital ring homomorphisms from `α` to `β`. -/ infixr:25 " →ₙ+* " => NonUnitalRingHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as a semigroup homomorphism `α →ₙ* β`. The `simp`-normal form is `(f : α →ₙ* β)`. -/ add_decl_doc NonUnitalRingHom.toMulHom #align non_unital_ring_hom.to_mul_hom NonUnitalRingHom.toMulHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc NonUnitalRingHom.toAddMonoidHom #align non_unital_ring_hom.to_add_monoid_hom NonUnitalRingHom.toAddMonoidHom section NonUnitalRingHomClass /-- `NonUnitalRingHomClass F α β` states that `F` is a type of non-unital (semi)ring homomorphisms. You should extend this class when you extend `NonUnitalRingHom`. -/ class NonUnitalRingHomClass (F : Type*) (α β : outParam (Type*)) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends MulHomClass F α β, AddMonoidHomClass F α β #align non_unital_ring_hom_class NonUnitalRingHomClass variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] [NonUnitalRingHomClass F α β] /-- Turn an element of a type `F` satisfying `NonUnitalRingHomClass F α β` into an actual `NonUnitalRingHom`. This is declared as the default coercion from `F` to `α →ₙ+* β`. -/ @[coe] def NonUnitalRingHomClass.toNonUnitalRingHom (f : F) : α →ₙ+* β := { (f : α →ₙ* β), (f : α →+ β) with } /-- Any type satisfying `NonUnitalRingHomClass` can be cast into `NonUnitalRingHom` via `NonUnitalRingHomClass.toNonUnitalRingHom`. -/ instance : CoeTC F (α →ₙ+* β) := ⟨NonUnitalRingHomClass.toNonUnitalRingHom⟩ end NonUnitalRingHomClass namespace NonUnitalRingHom section coe variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] instance : NonUnitalRingHomClass (α →ₙ+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := NonUnitalRingHom.map_add' map_zero := NonUnitalRingHom.map_zero' map_mul f := f.map_mul' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -/ -- instance : CoeFun (α →ₙ+* β) fun _ => α → β := -- ⟨fun f => f.toFun⟩ -- Porting note: removed due to new `coe` in Lean4 #noalign non_unital_ring_hom.to_fun_eq_coe #noalign non_unital_ring_hom.coe_mk #noalign non_unital_ring_hom.coe_coe initialize_simps_projections NonUnitalRingHom (toFun → apply) @[simp] theorem coe_toMulHom (f : α →ₙ+* β) : ⇑f.toMulHom = f := rfl #align non_unital_ring_hom.coe_to_mul_hom NonUnitalRingHom.coe_toMulHom @[simp] theorem coe_mulHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →ₙ* β) = ⟨f, h₁⟩ := rfl #align non_unital_ring_hom.coe_mul_hom_mk NonUnitalRingHom.coe_mulHom_mk theorem coe_toAddMonoidHom (f : α →ₙ+* β) : ⇑f.toAddMonoidHom = f := rfl #align non_unital_ring_hom.coe_to_add_monoid_hom NonUnitalRingHom.coe_toAddMonoidHom @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →+ β) = ⟨⟨f, h₂⟩, h₃⟩ := rfl #align non_unital_ring_hom.coe_add_monoid_hom_mk NonUnitalRingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : α →ₙ+* β := { f.toMulHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align non_unital_ring_hom.copy NonUnitalRingHom.copy @[simp] theorem coe_copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align non_unital_ring_hom.coe_copy NonUnitalRingHom.coe_copy theorem copy_eq (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align non_unital_ring_hom.copy_eq NonUnitalRingHom.copy_eq end coe section variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] variable (f : α →ₙ+* β) {x y : α} @[ext] theorem ext ⦃f g : α →ₙ+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align non_unital_ring_hom.ext NonUnitalRingHom.ext theorem ext_iff {f g : α →ₙ+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align non_unital_ring_hom.ext_iff NonUnitalRingHom.ext_iff @[simp] theorem mk_coe (f : α →ₙ+* β) (h₁ h₂ h₃) : NonUnitalRingHom.mk (MulHom.mk f h₁) h₂ h₃ = f := ext fun _ => rfl #align non_unital_ring_hom.mk_coe NonUnitalRingHom.mk_coe theorem coe_addMonoidHom_injective : Injective fun f : α →ₙ+* β => (f : α →+ β) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align non_unital_ring_hom.coe_add_monoid_hom_injective NonUnitalRingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_mulHom_injective : Injective fun f : α →ₙ+* β => (f : α →ₙ* β) := fun _ _ h => ext <| MulHom.congr_fun h #align non_unital_ring_hom.coe_mul_hom_injective NonUnitalRingHom.coe_mulHom_injective end variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/ protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by refine' { toFun := id.. } <;> intros <;> rfl #align non_unital_ring_hom.id NonUnitalRingHom.id instance : Zero (α →ₙ+* β) := ⟨{ toFun := 0, map_mul' := fun _ _ => (mul_zero (0 : β)).symm, map_zero' := rfl, map_add' := fun _ _ => (add_zero (0 : β)).symm }⟩ instance : Inhabited (α →ₙ+* β) := ⟨0⟩ @[simp] theorem coe_zero : ⇑(0 : α →ₙ+* β) = 0 := rfl #align non_unital_ring_hom.coe_zero NonUnitalRingHom.coe_zero @[simp] theorem zero_apply (x : α) : (0 : α →ₙ+* β) x = 0 := rfl #align non_unital_ring_hom.zero_apply NonUnitalRingHom.zero_apply @[simp] theorem id_apply (x : α) : NonUnitalRingHom.id α x = x := rfl #align non_unital_ring_hom.id_apply NonUnitalRingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (NonUnitalRingHom.id α : α →+ α) = AddMonoidHom.id α := rfl #align non_unital_ring_hom.coe_add_monoid_hom_id NonUnitalRingHom.coe_addMonoidHom_id @[simp] theorem coe_mulHom_id : (NonUnitalRingHom.id α : α →ₙ* α) = MulHom.id α := rfl #align non_unital_ring_hom.coe_mul_hom_id NonUnitalRingHom.coe_mulHom_id variable [NonUnitalNonAssocSemiring γ] /-- Composition of non-unital ring homomorphisms is a non-unital ring homomorphism. -/ def comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : α →ₙ+* γ := { g.toMulHom.comp f.toMulHom, g.toAddMonoidHom.comp f.toAddMonoidHom with } #align non_unital_ring_hom.comp NonUnitalRingHom.comp /-- Composition of non-unital ring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonUnitalNonAssocSemiring δ} (f : α →ₙ+* β) (g : β →ₙ+* γ) (h : γ →ₙ+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align non_unital_ring_hom.comp_assoc NonUnitalRingHom.comp_assoc @[simp] theorem coe_comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : ⇑(g.comp f) = g ∘ f := rfl #align non_unital_ring_hom.coe_comp NonUnitalRingHom.coe_comp @[simp] theorem comp_apply (g : β →ₙ+* γ) (f : α →ₙ+* β) (x : α) : g.comp f x = g (f x) := rfl #align non_unital_ring_hom.comp_apply NonUnitalRingHom.comp_apply variable (g : β →ₙ+* γ) (f : α →ₙ+* β) @[simp] theorem coe_comp_addMonoidHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : AddMonoidHom.mk ⟨g ∘ f, (g.comp f).map_zero'⟩ (g.comp f).map_add' = (g : β →+ γ).comp f := rfl #align non_unital_ring_hom.coe_comp_add_monoid_hom NonUnitalRingHom.coe_comp_addMonoidHom @[simp] theorem coe_comp_mulHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : MulHom.mk (g ∘ f) (g.comp f).map_mul' = (g : β →ₙ* γ).comp f := rfl #align non_unital_ring_hom.coe_comp_mul_hom NonUnitalRingHom.coe_comp_mulHom @[simp] theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 := by ext simp #align non_unital_ring_hom.comp_zero NonUnitalRingHom.comp_zero @[simp] theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 := by ext rfl #align non_unital_ring_hom.zero_comp NonUnitalRingHom.zero_comp @[simp] theorem comp_id (f : α →ₙ+* β) : f.comp (NonUnitalRingHom.id α) = f := ext fun _ => rfl #align non_unital_ring_hom.comp_id NonUnitalRingHom.comp_id @[simp] theorem id_comp (f : α →ₙ+* β) : (NonUnitalRingHom.id β).comp f = f := ext fun _ => rfl #align non_unital_ring_hom.id_comp NonUnitalRingHom.id_comp instance : MonoidWithZero (α →ₙ+* α) where one := NonUnitalRingHom.id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ zero := 0 mul_zero := comp_zero zero_mul := zero_comp theorem one_def : (1 : α →ₙ+* α) = NonUnitalRingHom.id α := rfl #align non_unital_ring_hom.one_def NonUnitalRingHom.one_def @[simp] theorem coe_one : ⇑(1 : α →ₙ+* α) = id := rfl #align non_unital_ring_hom.coe_one NonUnitalRingHom.coe_one theorem mul_def (f g : α →ₙ+* α) : f * g = f.comp g := rfl #align non_unital_ring_hom.mul_def NonUnitalRingHom.mul_def @[simp] theorem coe_mul (f g : α →ₙ+* α) : ⇑(f * g) = f ∘ g := rfl #align non_unital_ring_hom.coe_mul NonUnitalRingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →ₙ+* γ} {f : α →ₙ+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_right NonUnitalRingHom.cancel_right @[simp] theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_left NonUnitalRingHom.cancel_left end NonUnitalRingHom /-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too. This extends from both `MonoidHom` and `MonoidWithZeroHom` in order to put the fields in a sensible order, even though `MonoidWithZeroHom` already extends `MonoidHom`. -/ structure RingHom (α : Type*) (β : Type*) [NonAssocSemiring α] [NonAssocSemiring β] extends α →* β, α →+ β, α →ₙ+* β, α →*₀ β #align ring_hom RingHom /-- `α →+* β` denotes the type of ring homomorphisms from `α` to `β`. -/ infixr:25 " →+* " => RingHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid with zero homomorphism `α →*₀ β`. The `simp`-normal form is `(f : α →*₀ β)`. -/ add_decl_doc RingHom.toMonoidWithZeroHom #align ring_hom.to_monoid_with_zero_hom RingHom.toMonoidWithZeroHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid homomorphism `α →* β`. The `simp`-normal form is `(f : α →* β)`. -/ add_decl_doc RingHom.toMonoidHom #align ring_hom.to_monoid_hom RingHom.toMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc RingHom.toAddMonoidHom #align ring_hom.to_add_monoid_hom RingHom.toAddMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a non-unital ring homomorphism `α →ₙ+* β`. The `simp`-normal form is `(f : α →ₙ+* β)`. -/ add_decl_doc RingHom.toNonUnitalRingHom #align ring_hom.to_non_unital_ring_hom RingHom.toNonUnitalRingHom section RingHomClass /-- `RingHomClass F α β` states that `F` is a type of (semi)ring homomorphisms. You should extend this class when you extend `RingHom`. This extends from both `MonoidHomClass` and `MonoidWithZeroHomClass` in order to put the fields in a sensible order, even though `MonoidWithZeroHomClass` already extends `MonoidHomClass`. -/ class RingHomClass (F : Type*) (α β : outParam (Type*)) [NonAssocSemiring α] [NonAssocSemiring β] extends MonoidHomClass F α β, AddMonoidHomClass F α β, MonoidWithZeroHomClass F α β #align ring_hom_class RingHomClass set_option linter.deprecated false in /-- Ring homomorphisms preserve `bit1`. -/ @[simp] lemma map_bit1 [NonAssocSemiring α] [NonAssocSemiring β] [RingHomClass F α β] (f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) := by simp [bit1] #align map_bit1 map_bit1 -- Porting note: marked `{}` rather than `[]` to prevent dangerous instances variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} [RingHomClass F α β] /-- Turn an element of a type `F` satisfying `RingHomClass F α β` into an actual `RingHom`. This is declared as the default coercion from `F` to `α →+* β`. -/ @[coe] def RingHomClass.toRingHom (f : F) : α →+* β := { (f : α →* β), (f : α →+ β) with } /-- Any type satisfying `RingHomClass` can be cast into `RingHom` via `RingHomClass.toRingHom`. -/ instance : CoeTC F (α →+* β) := ⟨RingHomClass.toRingHom⟩ instance (priority := 100) RingHomClass.toNonUnitalRingHomClass : NonUnitalRingHomClass F α β := { ‹RingHomClass F α β› with } #align ring_hom_class.to_non_unital_ring_hom_class RingHomClass.toNonUnitalRingHomClass end RingHomClass namespace RingHom section coe /-! Throughout this section, some `Semiring` arguments are specified with `{}` instead of `[]`. See note [implicit instance arguments]. -/ variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} instance instRingHomClass : RingHomClass (α →+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := RingHom.map_add' map_zero := RingHom.map_zero' map_mul f := f.map_mul' map_one f := f.map_one' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -- -/ -- instance : CoeFun (α →+* β) fun _ => α → β := -- ⟨RingHom.toFun⟩ initialize_simps_projections RingHom (toFun → apply) -- Porting note: is this lemma still needed in Lean4? -- Porting note: because `f.toFun` really means `f.toMonoidHom.toOneHom.toFun` and -- `toMonoidHom_eq_coe` wants to simplify `f.toMonoidHom` to `(↑f : M →* N)`, this can't -- be a simp lemma anymore -- @[simp] theorem toFun_eq_coe (f : α →+* β) : f.toFun = f := rfl #align ring_hom.to_fun_eq_coe RingHom.toFun_eq_coe @[simp] theorem coe_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α → β) = f := rfl #align ring_hom.coe_mk RingHom.coe_mk @[simp] theorem coe_coe {F : Type*} [RingHomClass F α β] (f : F) : ((f : α →+* β) : α → β) = f := rfl #align ring_hom.coe_coe RingHom.coe_coe attribute [coe] RingHom.toMonoidHom instance coeToMonoidHom : Coe (α →+* β) (α →* β) := ⟨RingHom.toMonoidHom⟩ #align ring_hom.has_coe_monoid_hom RingHom.coeToMonoidHom -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_monoid_hom @[simp] theorem toMonoidHom_eq_coe (f : α →+* β) : f.toMonoidHom = f := rfl #align ring_hom.to_monoid_hom_eq_coe RingHom.toMonoidHom_eq_coe -- Porting note: this can't be a simp lemma anymore -- @[simp] theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f := by rfl #align ring_hom.to_monoid_with_zero_hom_eq_coe RingHom.toMonoidWithZeroHom_eq_coe @[simp] theorem coe_monoidHom_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α →* β) = f := rfl #align ring_hom.coe_monoid_hom_mk RingHom.coe_monoidHom_mk -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_add_monoid_hom @[simp] theorem toAddMonoidHom_eq_coe (f : α →+* β) : f.toAddMonoidHom = f := rfl #align ring_hom.to_add_monoid_hom_eq_coe RingHom.toAddMonoidHom_eq_coe @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃ h₄) : ((⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨⟨f, h₃⟩, h₄⟩ := rfl #align ring_hom.coe_add_monoid_hom_mk RingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ def copy (f : α →+* β) (f' : α → β) (h : f' = f) : α →+* β := { f.toMonoidWithZeroHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align ring_hom.copy RingHom.copy @[simp] theorem coe_copy (f : α →+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align ring_hom.coe_copy RingHom.coe_copy theorem copy_eq (f : α →+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align ring_hom.copy_eq RingHom.copy_eq end coe section variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} (f : α →+* β) {x y : α} theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x := FunLike.congr_fun h x #align ring_hom.congr_fun RingHom.congr_fun theorem congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y := FunLike.congr_arg f h #align ring_hom.congr_arg RingHom.congr_arg theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g := FunLike.coe_injective h #align ring_hom.coe_inj RingHom.coe_inj @[ext] theorem ext ⦃f g : α →+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align ring_hom.ext RingHom.ext theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align ring_hom.ext_iff RingHom.ext_iff @[simp] theorem mk_coe (f : α →+* β) (h₁ h₂ h₃ h₄) : RingHom.mk ⟨⟨f, h₁⟩, h₂⟩ h₃ h₄ = f := ext fun _ => rfl #align ring_hom.mk_coe RingHom.mk_coe theorem coe_addMonoidHom_injective : Injective (fun f : α →+* β => (f : α →+ β)) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align ring_hom.coe_add_monoid_hom_injective RingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_monoidHom_injective : Injective (fun f : α →+* β => (f : α →* β)) := fun _ _ h => ext <| MonoidHom.congr_fun h #align ring_hom.coe_monoid_hom_injective RingHom.coe_monoidHom_injective /-- Ring homomorphisms map zero to zero. -/ protected theorem map_zero (f : α →+* β) : f 0 = 0 := map_zero f #align ring_hom.map_zero RingHom.map_zero /-- Ring homomorphisms map one to one. -/ protected theorem map_one (f : α →+* β) : f 1 = 1 := map_one f #align ring_hom.map_one RingHom.map_one /-- Ring homomorphisms preserve addition. -/ protected theorem map_add (f : α →+* β) : ∀ a b, f (a + b) = f a + f b := map_add f #align ring_hom.map_add RingHom.map_add /-- Ring homomorphisms preserve multiplication. -/ protected theorem map_mul (f : α →+* β) : ∀ a b, f (a * b) = f a * f b := map_mul f #align ring_hom.map_mul RingHom.map_mul @[simp] theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 0 1) = ite p 0 1 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_zero_one RingHom.map_ite_zero_one @[simp] theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 1 0) = ite p 1 0 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_one_zero RingHom.map_ite_one_zero /-- `f : α →+* β` has a trivial codomain iff `f 1 = 0`. -/ theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 := by rw [map_one, eq_comm] #align ring_hom.codomain_trivial_iff_map_one_eq_zero RingHom.codomain_trivial_iff_map_one_eq_zero /-- `f : α →+* β` has a trivial codomain iff it has a trivial range. -/ theorem codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 := f.codomain_trivial_iff_map_one_eq_zero.trans ⟨fun h x => by rw [← mul_one x, map_mul, h, mul_zero], fun h => h 1⟩ #align ring_hom.codomain_trivial_iff_range_trivial RingHom.codomain_trivial_iff_range_trivial /-- `f : α →+* β` doesn't map `1` to `0` if `β` is nontrivial -/ theorem map_one_ne_zero [Nontrivial β] : f 1 ≠ 0 := mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one #align ring_hom.map_one_ne_zero RingHom.map_one_ne_zero /-- If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. -/ theorem domain_nontrivial [Nontrivial β] : Nontrivial α := ⟨⟨1, 0, mt (fun h => show f 1 = 0 by rw [h, map_zero]) f.map_one_ne_zero⟩⟩ #align ring_hom.domain_nontrivial RingHom.domain_nontrivial theorem codomain_trivial (f : α →+* β) [h : Subsingleton α] : Subsingleton β := (subsingleton_or_nontrivial β).resolve_right fun _ => not_nontrivial_iff_subsingleton.mpr h f.domain_nontrivial #align ring_hom.codomain_trivial RingHom.codomain_trivial end /-- Ring homomorphisms preserve additive inverse. -/ protected theorem map_neg [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x : α) : f (-x) = -f x := map_neg f x #align ring_hom.map_neg RingHom.map_neg /-- Ring homomorphisms preserve subtraction. -/ protected theorem map_sub [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x y : α) : f (x - y) = f x - f y := map_sub f x y #align ring_hom.map_sub RingHom.map_sub /-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/ def mk' [NonAssocSemiring α] [NonAssocRing β] (f : α →* β) (map_add : ∀ a b, f (a + b) = f a + f b) : α →+* β := { AddMonoidHom.mk' f map_add, f with } #align ring_hom.mk' RingHom.mk' variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} /-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;>

intros

/-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;>

Mathlib.Algebra.Ring.Hom.Defs.631_0.KyHvVYrIs9pW9ZQ

/-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α

Mathlib_Algebra_Ring_Hom_Defs

case refine'_1 F : Type u_1 α✝ : Type u_2 β : Type u_3 γ : Type u_4 x✝¹ : NonAssocSemiring α✝ x✝ : NonAssocSemiring β α : Type u_5 inst✝ : NonAssocSemiring α ⊢ _root_.id 1 = 1

/- Copyright (c) 2019 Amelia Livingston. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Amelia Livingston, Jireh Loreaux -/ import Mathlib.Algebra.Ring.Defs import Mathlib.Algebra.Ring.Basic import Mathlib.Data.Pi.Algebra #align_import algebra.hom.ring from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9" /-! # Homomorphisms of semirings and rings This file defines bundled homomorphisms of (non-unital) semirings and rings. As with monoid and groups, we use the same structure `RingHom a β`, a.k.a. `α →+* β`, for both types of homomorphisms. ## Main definitions * `NonUnitalRingHom`: Non-unital (semi)ring homomorphisms. Additive monoid homomorphism which preserve multiplication. * `RingHom`: (Semi)ring homomorphisms. Monoid homomorphisms which are also additive monoid homomorphism. ## Notations * `→ₙ+*`: Non-unital (semi)ring homs * `→+*`: (Semi)ring homs ## Implementation notes * There's a coercion from bundled homs to fun, and the canonical notation is to use the bundled hom as a function via this coercion. * There is no `SemiringHom` -- the idea is that `RingHom` is used. The constructor for a `RingHom` between semirings needs a proof of `map_zero`, `map_one` and `map_add` as well as `map_mul`; a separate constructor `RingHom.mk'` will construct ring homs between rings from monoid homs given only a proof that addition is preserved. ## Tags `RingHom`, `SemiringHom` -/ open Function variable {F α β γ : Type*} /-- Bundled non-unital semiring homomorphisms `α →ₙ+* β`; use this for bundled non-unital ring homomorphisms too. When possible, instead of parametrizing results over `(f : α →ₙ+* β)`, you should parametrize over `(F : Type*) [NonUnitalRingHomClass F α β] (f : F)`. When you extend this structure, make sure to extend `NonUnitalRingHomClass`. -/ structure NonUnitalRingHom (α β : Type*) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends α →ₙ* β, α →+ β #align non_unital_ring_hom NonUnitalRingHom /-- `α →ₙ+* β` denotes the type of non-unital ring homomorphisms from `α` to `β`. -/ infixr:25 " →ₙ+* " => NonUnitalRingHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as a semigroup homomorphism `α →ₙ* β`. The `simp`-normal form is `(f : α →ₙ* β)`. -/ add_decl_doc NonUnitalRingHom.toMulHom #align non_unital_ring_hom.to_mul_hom NonUnitalRingHom.toMulHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc NonUnitalRingHom.toAddMonoidHom #align non_unital_ring_hom.to_add_monoid_hom NonUnitalRingHom.toAddMonoidHom section NonUnitalRingHomClass /-- `NonUnitalRingHomClass F α β` states that `F` is a type of non-unital (semi)ring homomorphisms. You should extend this class when you extend `NonUnitalRingHom`. -/ class NonUnitalRingHomClass (F : Type*) (α β : outParam (Type*)) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends MulHomClass F α β, AddMonoidHomClass F α β #align non_unital_ring_hom_class NonUnitalRingHomClass variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] [NonUnitalRingHomClass F α β] /-- Turn an element of a type `F` satisfying `NonUnitalRingHomClass F α β` into an actual `NonUnitalRingHom`. This is declared as the default coercion from `F` to `α →ₙ+* β`. -/ @[coe] def NonUnitalRingHomClass.toNonUnitalRingHom (f : F) : α →ₙ+* β := { (f : α →ₙ* β), (f : α →+ β) with } /-- Any type satisfying `NonUnitalRingHomClass` can be cast into `NonUnitalRingHom` via `NonUnitalRingHomClass.toNonUnitalRingHom`. -/ instance : CoeTC F (α →ₙ+* β) := ⟨NonUnitalRingHomClass.toNonUnitalRingHom⟩ end NonUnitalRingHomClass namespace NonUnitalRingHom section coe variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] instance : NonUnitalRingHomClass (α →ₙ+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := NonUnitalRingHom.map_add' map_zero := NonUnitalRingHom.map_zero' map_mul f := f.map_mul' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -/ -- instance : CoeFun (α →ₙ+* β) fun _ => α → β := -- ⟨fun f => f.toFun⟩ -- Porting note: removed due to new `coe` in Lean4 #noalign non_unital_ring_hom.to_fun_eq_coe #noalign non_unital_ring_hom.coe_mk #noalign non_unital_ring_hom.coe_coe initialize_simps_projections NonUnitalRingHom (toFun → apply) @[simp] theorem coe_toMulHom (f : α →ₙ+* β) : ⇑f.toMulHom = f := rfl #align non_unital_ring_hom.coe_to_mul_hom NonUnitalRingHom.coe_toMulHom @[simp] theorem coe_mulHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →ₙ* β) = ⟨f, h₁⟩ := rfl #align non_unital_ring_hom.coe_mul_hom_mk NonUnitalRingHom.coe_mulHom_mk theorem coe_toAddMonoidHom (f : α →ₙ+* β) : ⇑f.toAddMonoidHom = f := rfl #align non_unital_ring_hom.coe_to_add_monoid_hom NonUnitalRingHom.coe_toAddMonoidHom @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →+ β) = ⟨⟨f, h₂⟩, h₃⟩ := rfl #align non_unital_ring_hom.coe_add_monoid_hom_mk NonUnitalRingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : α →ₙ+* β := { f.toMulHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align non_unital_ring_hom.copy NonUnitalRingHom.copy @[simp] theorem coe_copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align non_unital_ring_hom.coe_copy NonUnitalRingHom.coe_copy theorem copy_eq (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align non_unital_ring_hom.copy_eq NonUnitalRingHom.copy_eq end coe section variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] variable (f : α →ₙ+* β) {x y : α} @[ext] theorem ext ⦃f g : α →ₙ+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align non_unital_ring_hom.ext NonUnitalRingHom.ext theorem ext_iff {f g : α →ₙ+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align non_unital_ring_hom.ext_iff NonUnitalRingHom.ext_iff @[simp] theorem mk_coe (f : α →ₙ+* β) (h₁ h₂ h₃) : NonUnitalRingHom.mk (MulHom.mk f h₁) h₂ h₃ = f := ext fun _ => rfl #align non_unital_ring_hom.mk_coe NonUnitalRingHom.mk_coe theorem coe_addMonoidHom_injective : Injective fun f : α →ₙ+* β => (f : α →+ β) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align non_unital_ring_hom.coe_add_monoid_hom_injective NonUnitalRingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_mulHom_injective : Injective fun f : α →ₙ+* β => (f : α →ₙ* β) := fun _ _ h => ext <| MulHom.congr_fun h #align non_unital_ring_hom.coe_mul_hom_injective NonUnitalRingHom.coe_mulHom_injective end variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/ protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by refine' { toFun := id.. } <;> intros <;> rfl #align non_unital_ring_hom.id NonUnitalRingHom.id instance : Zero (α →ₙ+* β) := ⟨{ toFun := 0, map_mul' := fun _ _ => (mul_zero (0 : β)).symm, map_zero' := rfl, map_add' := fun _ _ => (add_zero (0 : β)).symm }⟩ instance : Inhabited (α →ₙ+* β) := ⟨0⟩ @[simp] theorem coe_zero : ⇑(0 : α →ₙ+* β) = 0 := rfl #align non_unital_ring_hom.coe_zero NonUnitalRingHom.coe_zero @[simp] theorem zero_apply (x : α) : (0 : α →ₙ+* β) x = 0 := rfl #align non_unital_ring_hom.zero_apply NonUnitalRingHom.zero_apply @[simp] theorem id_apply (x : α) : NonUnitalRingHom.id α x = x := rfl #align non_unital_ring_hom.id_apply NonUnitalRingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (NonUnitalRingHom.id α : α →+ α) = AddMonoidHom.id α := rfl #align non_unital_ring_hom.coe_add_monoid_hom_id NonUnitalRingHom.coe_addMonoidHom_id @[simp] theorem coe_mulHom_id : (NonUnitalRingHom.id α : α →ₙ* α) = MulHom.id α := rfl #align non_unital_ring_hom.coe_mul_hom_id NonUnitalRingHom.coe_mulHom_id variable [NonUnitalNonAssocSemiring γ] /-- Composition of non-unital ring homomorphisms is a non-unital ring homomorphism. -/ def comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : α →ₙ+* γ := { g.toMulHom.comp f.toMulHom, g.toAddMonoidHom.comp f.toAddMonoidHom with } #align non_unital_ring_hom.comp NonUnitalRingHom.comp /-- Composition of non-unital ring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonUnitalNonAssocSemiring δ} (f : α →ₙ+* β) (g : β →ₙ+* γ) (h : γ →ₙ+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align non_unital_ring_hom.comp_assoc NonUnitalRingHom.comp_assoc @[simp] theorem coe_comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : ⇑(g.comp f) = g ∘ f := rfl #align non_unital_ring_hom.coe_comp NonUnitalRingHom.coe_comp @[simp] theorem comp_apply (g : β →ₙ+* γ) (f : α →ₙ+* β) (x : α) : g.comp f x = g (f x) := rfl #align non_unital_ring_hom.comp_apply NonUnitalRingHom.comp_apply variable (g : β →ₙ+* γ) (f : α →ₙ+* β) @[simp] theorem coe_comp_addMonoidHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : AddMonoidHom.mk ⟨g ∘ f, (g.comp f).map_zero'⟩ (g.comp f).map_add' = (g : β →+ γ).comp f := rfl #align non_unital_ring_hom.coe_comp_add_monoid_hom NonUnitalRingHom.coe_comp_addMonoidHom @[simp] theorem coe_comp_mulHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : MulHom.mk (g ∘ f) (g.comp f).map_mul' = (g : β →ₙ* γ).comp f := rfl #align non_unital_ring_hom.coe_comp_mul_hom NonUnitalRingHom.coe_comp_mulHom @[simp] theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 := by ext simp #align non_unital_ring_hom.comp_zero NonUnitalRingHom.comp_zero @[simp] theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 := by ext rfl #align non_unital_ring_hom.zero_comp NonUnitalRingHom.zero_comp @[simp] theorem comp_id (f : α →ₙ+* β) : f.comp (NonUnitalRingHom.id α) = f := ext fun _ => rfl #align non_unital_ring_hom.comp_id NonUnitalRingHom.comp_id @[simp] theorem id_comp (f : α →ₙ+* β) : (NonUnitalRingHom.id β).comp f = f := ext fun _ => rfl #align non_unital_ring_hom.id_comp NonUnitalRingHom.id_comp instance : MonoidWithZero (α →ₙ+* α) where one := NonUnitalRingHom.id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ zero := 0 mul_zero := comp_zero zero_mul := zero_comp theorem one_def : (1 : α →ₙ+* α) = NonUnitalRingHom.id α := rfl #align non_unital_ring_hom.one_def NonUnitalRingHom.one_def @[simp] theorem coe_one : ⇑(1 : α →ₙ+* α) = id := rfl #align non_unital_ring_hom.coe_one NonUnitalRingHom.coe_one theorem mul_def (f g : α →ₙ+* α) : f * g = f.comp g := rfl #align non_unital_ring_hom.mul_def NonUnitalRingHom.mul_def @[simp] theorem coe_mul (f g : α →ₙ+* α) : ⇑(f * g) = f ∘ g := rfl #align non_unital_ring_hom.coe_mul NonUnitalRingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →ₙ+* γ} {f : α →ₙ+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_right NonUnitalRingHom.cancel_right @[simp] theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_left NonUnitalRingHom.cancel_left end NonUnitalRingHom /-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too. This extends from both `MonoidHom` and `MonoidWithZeroHom` in order to put the fields in a sensible order, even though `MonoidWithZeroHom` already extends `MonoidHom`. -/ structure RingHom (α : Type*) (β : Type*) [NonAssocSemiring α] [NonAssocSemiring β] extends α →* β, α →+ β, α →ₙ+* β, α →*₀ β #align ring_hom RingHom /-- `α →+* β` denotes the type of ring homomorphisms from `α` to `β`. -/ infixr:25 " →+* " => RingHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid with zero homomorphism `α →*₀ β`. The `simp`-normal form is `(f : α →*₀ β)`. -/ add_decl_doc RingHom.toMonoidWithZeroHom #align ring_hom.to_monoid_with_zero_hom RingHom.toMonoidWithZeroHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid homomorphism `α →* β`. The `simp`-normal form is `(f : α →* β)`. -/ add_decl_doc RingHom.toMonoidHom #align ring_hom.to_monoid_hom RingHom.toMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc RingHom.toAddMonoidHom #align ring_hom.to_add_monoid_hom RingHom.toAddMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a non-unital ring homomorphism `α →ₙ+* β`. The `simp`-normal form is `(f : α →ₙ+* β)`. -/ add_decl_doc RingHom.toNonUnitalRingHom #align ring_hom.to_non_unital_ring_hom RingHom.toNonUnitalRingHom section RingHomClass /-- `RingHomClass F α β` states that `F` is a type of (semi)ring homomorphisms. You should extend this class when you extend `RingHom`. This extends from both `MonoidHomClass` and `MonoidWithZeroHomClass` in order to put the fields in a sensible order, even though `MonoidWithZeroHomClass` already extends `MonoidHomClass`. -/ class RingHomClass (F : Type*) (α β : outParam (Type*)) [NonAssocSemiring α] [NonAssocSemiring β] extends MonoidHomClass F α β, AddMonoidHomClass F α β, MonoidWithZeroHomClass F α β #align ring_hom_class RingHomClass set_option linter.deprecated false in /-- Ring homomorphisms preserve `bit1`. -/ @[simp] lemma map_bit1 [NonAssocSemiring α] [NonAssocSemiring β] [RingHomClass F α β] (f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) := by simp [bit1] #align map_bit1 map_bit1 -- Porting note: marked `{}` rather than `[]` to prevent dangerous instances variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} [RingHomClass F α β] /-- Turn an element of a type `F` satisfying `RingHomClass F α β` into an actual `RingHom`. This is declared as the default coercion from `F` to `α →+* β`. -/ @[coe] def RingHomClass.toRingHom (f : F) : α →+* β := { (f : α →* β), (f : α →+ β) with } /-- Any type satisfying `RingHomClass` can be cast into `RingHom` via `RingHomClass.toRingHom`. -/ instance : CoeTC F (α →+* β) := ⟨RingHomClass.toRingHom⟩ instance (priority := 100) RingHomClass.toNonUnitalRingHomClass : NonUnitalRingHomClass F α β := { ‹RingHomClass F α β› with } #align ring_hom_class.to_non_unital_ring_hom_class RingHomClass.toNonUnitalRingHomClass end RingHomClass namespace RingHom section coe /-! Throughout this section, some `Semiring` arguments are specified with `{}` instead of `[]`. See note [implicit instance arguments]. -/ variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} instance instRingHomClass : RingHomClass (α →+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := RingHom.map_add' map_zero := RingHom.map_zero' map_mul f := f.map_mul' map_one f := f.map_one' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -- -/ -- instance : CoeFun (α →+* β) fun _ => α → β := -- ⟨RingHom.toFun⟩ initialize_simps_projections RingHom (toFun → apply) -- Porting note: is this lemma still needed in Lean4? -- Porting note: because `f.toFun` really means `f.toMonoidHom.toOneHom.toFun` and -- `toMonoidHom_eq_coe` wants to simplify `f.toMonoidHom` to `(↑f : M →* N)`, this can't -- be a simp lemma anymore -- @[simp] theorem toFun_eq_coe (f : α →+* β) : f.toFun = f := rfl #align ring_hom.to_fun_eq_coe RingHom.toFun_eq_coe @[simp] theorem coe_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α → β) = f := rfl #align ring_hom.coe_mk RingHom.coe_mk @[simp] theorem coe_coe {F : Type*} [RingHomClass F α β] (f : F) : ((f : α →+* β) : α → β) = f := rfl #align ring_hom.coe_coe RingHom.coe_coe attribute [coe] RingHom.toMonoidHom instance coeToMonoidHom : Coe (α →+* β) (α →* β) := ⟨RingHom.toMonoidHom⟩ #align ring_hom.has_coe_monoid_hom RingHom.coeToMonoidHom -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_monoid_hom @[simp] theorem toMonoidHom_eq_coe (f : α →+* β) : f.toMonoidHom = f := rfl #align ring_hom.to_monoid_hom_eq_coe RingHom.toMonoidHom_eq_coe -- Porting note: this can't be a simp lemma anymore -- @[simp] theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f := by rfl #align ring_hom.to_monoid_with_zero_hom_eq_coe RingHom.toMonoidWithZeroHom_eq_coe @[simp] theorem coe_monoidHom_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α →* β) = f := rfl #align ring_hom.coe_monoid_hom_mk RingHom.coe_monoidHom_mk -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_add_monoid_hom @[simp] theorem toAddMonoidHom_eq_coe (f : α →+* β) : f.toAddMonoidHom = f := rfl #align ring_hom.to_add_monoid_hom_eq_coe RingHom.toAddMonoidHom_eq_coe @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃ h₄) : ((⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨⟨f, h₃⟩, h₄⟩ := rfl #align ring_hom.coe_add_monoid_hom_mk RingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ def copy (f : α →+* β) (f' : α → β) (h : f' = f) : α →+* β := { f.toMonoidWithZeroHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align ring_hom.copy RingHom.copy @[simp] theorem coe_copy (f : α →+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align ring_hom.coe_copy RingHom.coe_copy theorem copy_eq (f : α →+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align ring_hom.copy_eq RingHom.copy_eq end coe section variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} (f : α →+* β) {x y : α} theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x := FunLike.congr_fun h x #align ring_hom.congr_fun RingHom.congr_fun theorem congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y := FunLike.congr_arg f h #align ring_hom.congr_arg RingHom.congr_arg theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g := FunLike.coe_injective h #align ring_hom.coe_inj RingHom.coe_inj @[ext] theorem ext ⦃f g : α →+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align ring_hom.ext RingHom.ext theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align ring_hom.ext_iff RingHom.ext_iff @[simp] theorem mk_coe (f : α →+* β) (h₁ h₂ h₃ h₄) : RingHom.mk ⟨⟨f, h₁⟩, h₂⟩ h₃ h₄ = f := ext fun _ => rfl #align ring_hom.mk_coe RingHom.mk_coe theorem coe_addMonoidHom_injective : Injective (fun f : α →+* β => (f : α →+ β)) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align ring_hom.coe_add_monoid_hom_injective RingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_monoidHom_injective : Injective (fun f : α →+* β => (f : α →* β)) := fun _ _ h => ext <| MonoidHom.congr_fun h #align ring_hom.coe_monoid_hom_injective RingHom.coe_monoidHom_injective /-- Ring homomorphisms map zero to zero. -/ protected theorem map_zero (f : α →+* β) : f 0 = 0 := map_zero f #align ring_hom.map_zero RingHom.map_zero /-- Ring homomorphisms map one to one. -/ protected theorem map_one (f : α →+* β) : f 1 = 1 := map_one f #align ring_hom.map_one RingHom.map_one /-- Ring homomorphisms preserve addition. -/ protected theorem map_add (f : α →+* β) : ∀ a b, f (a + b) = f a + f b := map_add f #align ring_hom.map_add RingHom.map_add /-- Ring homomorphisms preserve multiplication. -/ protected theorem map_mul (f : α →+* β) : ∀ a b, f (a * b) = f a * f b := map_mul f #align ring_hom.map_mul RingHom.map_mul @[simp] theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 0 1) = ite p 0 1 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_zero_one RingHom.map_ite_zero_one @[simp] theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 1 0) = ite p 1 0 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_one_zero RingHom.map_ite_one_zero /-- `f : α →+* β` has a trivial codomain iff `f 1 = 0`. -/ theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 := by rw [map_one, eq_comm] #align ring_hom.codomain_trivial_iff_map_one_eq_zero RingHom.codomain_trivial_iff_map_one_eq_zero /-- `f : α →+* β` has a trivial codomain iff it has a trivial range. -/ theorem codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 := f.codomain_trivial_iff_map_one_eq_zero.trans ⟨fun h x => by rw [← mul_one x, map_mul, h, mul_zero], fun h => h 1⟩ #align ring_hom.codomain_trivial_iff_range_trivial RingHom.codomain_trivial_iff_range_trivial /-- `f : α →+* β` doesn't map `1` to `0` if `β` is nontrivial -/ theorem map_one_ne_zero [Nontrivial β] : f 1 ≠ 0 := mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one #align ring_hom.map_one_ne_zero RingHom.map_one_ne_zero /-- If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. -/ theorem domain_nontrivial [Nontrivial β] : Nontrivial α := ⟨⟨1, 0, mt (fun h => show f 1 = 0 by rw [h, map_zero]) f.map_one_ne_zero⟩⟩ #align ring_hom.domain_nontrivial RingHom.domain_nontrivial theorem codomain_trivial (f : α →+* β) [h : Subsingleton α] : Subsingleton β := (subsingleton_or_nontrivial β).resolve_right fun _ => not_nontrivial_iff_subsingleton.mpr h f.domain_nontrivial #align ring_hom.codomain_trivial RingHom.codomain_trivial end /-- Ring homomorphisms preserve additive inverse. -/ protected theorem map_neg [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x : α) : f (-x) = -f x := map_neg f x #align ring_hom.map_neg RingHom.map_neg /-- Ring homomorphisms preserve subtraction. -/ protected theorem map_sub [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x y : α) : f (x - y) = f x - f y := map_sub f x y #align ring_hom.map_sub RingHom.map_sub /-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/ def mk' [NonAssocSemiring α] [NonAssocRing β] (f : α →* β) (map_add : ∀ a b, f (a + b) = f a + f b) : α →+* β := { AddMonoidHom.mk' f map_add, f with } #align ring_hom.mk' RingHom.mk' variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} /-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;> intros <;>

rfl

/-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;> intros <;>

Mathlib.Algebra.Ring.Hom.Defs.631_0.KyHvVYrIs9pW9ZQ

/-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α

Mathlib_Algebra_Ring_Hom_Defs

case refine'_2 F : Type u_1 α✝ : Type u_2 β : Type u_3 γ : Type u_4 x✝² : NonAssocSemiring α✝ x✝¹ : NonAssocSemiring β α : Type u_5 inst✝ : NonAssocSemiring α x✝ y✝ : α ⊢ OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } (x✝ * y✝) = OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } x✝ * OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } y✝

/- Copyright (c) 2019 Amelia Livingston. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Amelia Livingston, Jireh Loreaux -/ import Mathlib.Algebra.Ring.Defs import Mathlib.Algebra.Ring.Basic import Mathlib.Data.Pi.Algebra #align_import algebra.hom.ring from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9" /-! # Homomorphisms of semirings and rings This file defines bundled homomorphisms of (non-unital) semirings and rings. As with monoid and groups, we use the same structure `RingHom a β`, a.k.a. `α →+* β`, for both types of homomorphisms. ## Main definitions * `NonUnitalRingHom`: Non-unital (semi)ring homomorphisms. Additive monoid homomorphism which preserve multiplication. * `RingHom`: (Semi)ring homomorphisms. Monoid homomorphisms which are also additive monoid homomorphism. ## Notations * `→ₙ+*`: Non-unital (semi)ring homs * `→+*`: (Semi)ring homs ## Implementation notes * There's a coercion from bundled homs to fun, and the canonical notation is to use the bundled hom as a function via this coercion. * There is no `SemiringHom` -- the idea is that `RingHom` is used. The constructor for a `RingHom` between semirings needs a proof of `map_zero`, `map_one` and `map_add` as well as `map_mul`; a separate constructor `RingHom.mk'` will construct ring homs between rings from monoid homs given only a proof that addition is preserved. ## Tags `RingHom`, `SemiringHom` -/ open Function variable {F α β γ : Type*} /-- Bundled non-unital semiring homomorphisms `α →ₙ+* β`; use this for bundled non-unital ring homomorphisms too. When possible, instead of parametrizing results over `(f : α →ₙ+* β)`, you should parametrize over `(F : Type*) [NonUnitalRingHomClass F α β] (f : F)`. When you extend this structure, make sure to extend `NonUnitalRingHomClass`. -/ structure NonUnitalRingHom (α β : Type*) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends α →ₙ* β, α →+ β #align non_unital_ring_hom NonUnitalRingHom /-- `α →ₙ+* β` denotes the type of non-unital ring homomorphisms from `α` to `β`. -/ infixr:25 " →ₙ+* " => NonUnitalRingHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as a semigroup homomorphism `α →ₙ* β`. The `simp`-normal form is `(f : α →ₙ* β)`. -/ add_decl_doc NonUnitalRingHom.toMulHom #align non_unital_ring_hom.to_mul_hom NonUnitalRingHom.toMulHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc NonUnitalRingHom.toAddMonoidHom #align non_unital_ring_hom.to_add_monoid_hom NonUnitalRingHom.toAddMonoidHom section NonUnitalRingHomClass /-- `NonUnitalRingHomClass F α β` states that `F` is a type of non-unital (semi)ring homomorphisms. You should extend this class when you extend `NonUnitalRingHom`. -/ class NonUnitalRingHomClass (F : Type*) (α β : outParam (Type*)) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends MulHomClass F α β, AddMonoidHomClass F α β #align non_unital_ring_hom_class NonUnitalRingHomClass variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] [NonUnitalRingHomClass F α β] /-- Turn an element of a type `F` satisfying `NonUnitalRingHomClass F α β` into an actual `NonUnitalRingHom`. This is declared as the default coercion from `F` to `α →ₙ+* β`. -/ @[coe] def NonUnitalRingHomClass.toNonUnitalRingHom (f : F) : α →ₙ+* β := { (f : α →ₙ* β), (f : α →+ β) with } /-- Any type satisfying `NonUnitalRingHomClass` can be cast into `NonUnitalRingHom` via `NonUnitalRingHomClass.toNonUnitalRingHom`. -/ instance : CoeTC F (α →ₙ+* β) := ⟨NonUnitalRingHomClass.toNonUnitalRingHom⟩ end NonUnitalRingHomClass namespace NonUnitalRingHom section coe variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] instance : NonUnitalRingHomClass (α →ₙ+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := NonUnitalRingHom.map_add' map_zero := NonUnitalRingHom.map_zero' map_mul f := f.map_mul' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -/ -- instance : CoeFun (α →ₙ+* β) fun _ => α → β := -- ⟨fun f => f.toFun⟩ -- Porting note: removed due to new `coe` in Lean4 #noalign non_unital_ring_hom.to_fun_eq_coe #noalign non_unital_ring_hom.coe_mk #noalign non_unital_ring_hom.coe_coe initialize_simps_projections NonUnitalRingHom (toFun → apply) @[simp] theorem coe_toMulHom (f : α →ₙ+* β) : ⇑f.toMulHom = f := rfl #align non_unital_ring_hom.coe_to_mul_hom NonUnitalRingHom.coe_toMulHom @[simp] theorem coe_mulHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →ₙ* β) = ⟨f, h₁⟩ := rfl #align non_unital_ring_hom.coe_mul_hom_mk NonUnitalRingHom.coe_mulHom_mk theorem coe_toAddMonoidHom (f : α →ₙ+* β) : ⇑f.toAddMonoidHom = f := rfl #align non_unital_ring_hom.coe_to_add_monoid_hom NonUnitalRingHom.coe_toAddMonoidHom @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →+ β) = ⟨⟨f, h₂⟩, h₃⟩ := rfl #align non_unital_ring_hom.coe_add_monoid_hom_mk NonUnitalRingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : α →ₙ+* β := { f.toMulHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align non_unital_ring_hom.copy NonUnitalRingHom.copy @[simp] theorem coe_copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align non_unital_ring_hom.coe_copy NonUnitalRingHom.coe_copy theorem copy_eq (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align non_unital_ring_hom.copy_eq NonUnitalRingHom.copy_eq end coe section variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] variable (f : α →ₙ+* β) {x y : α} @[ext] theorem ext ⦃f g : α →ₙ+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align non_unital_ring_hom.ext NonUnitalRingHom.ext theorem ext_iff {f g : α →ₙ+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align non_unital_ring_hom.ext_iff NonUnitalRingHom.ext_iff @[simp] theorem mk_coe (f : α →ₙ+* β) (h₁ h₂ h₃) : NonUnitalRingHom.mk (MulHom.mk f h₁) h₂ h₃ = f := ext fun _ => rfl #align non_unital_ring_hom.mk_coe NonUnitalRingHom.mk_coe theorem coe_addMonoidHom_injective : Injective fun f : α →ₙ+* β => (f : α →+ β) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align non_unital_ring_hom.coe_add_monoid_hom_injective NonUnitalRingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_mulHom_injective : Injective fun f : α →ₙ+* β => (f : α →ₙ* β) := fun _ _ h => ext <| MulHom.congr_fun h #align non_unital_ring_hom.coe_mul_hom_injective NonUnitalRingHom.coe_mulHom_injective end variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/ protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by refine' { toFun := id.. } <;> intros <;> rfl #align non_unital_ring_hom.id NonUnitalRingHom.id instance : Zero (α →ₙ+* β) := ⟨{ toFun := 0, map_mul' := fun _ _ => (mul_zero (0 : β)).symm, map_zero' := rfl, map_add' := fun _ _ => (add_zero (0 : β)).symm }⟩ instance : Inhabited (α →ₙ+* β) := ⟨0⟩ @[simp] theorem coe_zero : ⇑(0 : α →ₙ+* β) = 0 := rfl #align non_unital_ring_hom.coe_zero NonUnitalRingHom.coe_zero @[simp] theorem zero_apply (x : α) : (0 : α →ₙ+* β) x = 0 := rfl #align non_unital_ring_hom.zero_apply NonUnitalRingHom.zero_apply @[simp] theorem id_apply (x : α) : NonUnitalRingHom.id α x = x := rfl #align non_unital_ring_hom.id_apply NonUnitalRingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (NonUnitalRingHom.id α : α →+ α) = AddMonoidHom.id α := rfl #align non_unital_ring_hom.coe_add_monoid_hom_id NonUnitalRingHom.coe_addMonoidHom_id @[simp] theorem coe_mulHom_id : (NonUnitalRingHom.id α : α →ₙ* α) = MulHom.id α := rfl #align non_unital_ring_hom.coe_mul_hom_id NonUnitalRingHom.coe_mulHom_id variable [NonUnitalNonAssocSemiring γ] /-- Composition of non-unital ring homomorphisms is a non-unital ring homomorphism. -/ def comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : α →ₙ+* γ := { g.toMulHom.comp f.toMulHom, g.toAddMonoidHom.comp f.toAddMonoidHom with } #align non_unital_ring_hom.comp NonUnitalRingHom.comp /-- Composition of non-unital ring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonUnitalNonAssocSemiring δ} (f : α →ₙ+* β) (g : β →ₙ+* γ) (h : γ →ₙ+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align non_unital_ring_hom.comp_assoc NonUnitalRingHom.comp_assoc @[simp] theorem coe_comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : ⇑(g.comp f) = g ∘ f := rfl #align non_unital_ring_hom.coe_comp NonUnitalRingHom.coe_comp @[simp] theorem comp_apply (g : β →ₙ+* γ) (f : α →ₙ+* β) (x : α) : g.comp f x = g (f x) := rfl #align non_unital_ring_hom.comp_apply NonUnitalRingHom.comp_apply variable (g : β →ₙ+* γ) (f : α →ₙ+* β) @[simp] theorem coe_comp_addMonoidHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : AddMonoidHom.mk ⟨g ∘ f, (g.comp f).map_zero'⟩ (g.comp f).map_add' = (g : β →+ γ).comp f := rfl #align non_unital_ring_hom.coe_comp_add_monoid_hom NonUnitalRingHom.coe_comp_addMonoidHom @[simp] theorem coe_comp_mulHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : MulHom.mk (g ∘ f) (g.comp f).map_mul' = (g : β →ₙ* γ).comp f := rfl #align non_unital_ring_hom.coe_comp_mul_hom NonUnitalRingHom.coe_comp_mulHom @[simp] theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 := by ext simp #align non_unital_ring_hom.comp_zero NonUnitalRingHom.comp_zero @[simp] theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 := by ext rfl #align non_unital_ring_hom.zero_comp NonUnitalRingHom.zero_comp @[simp] theorem comp_id (f : α →ₙ+* β) : f.comp (NonUnitalRingHom.id α) = f := ext fun _ => rfl #align non_unital_ring_hom.comp_id NonUnitalRingHom.comp_id @[simp] theorem id_comp (f : α →ₙ+* β) : (NonUnitalRingHom.id β).comp f = f := ext fun _ => rfl #align non_unital_ring_hom.id_comp NonUnitalRingHom.id_comp instance : MonoidWithZero (α →ₙ+* α) where one := NonUnitalRingHom.id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ zero := 0 mul_zero := comp_zero zero_mul := zero_comp theorem one_def : (1 : α →ₙ+* α) = NonUnitalRingHom.id α := rfl #align non_unital_ring_hom.one_def NonUnitalRingHom.one_def @[simp] theorem coe_one : ⇑(1 : α →ₙ+* α) = id := rfl #align non_unital_ring_hom.coe_one NonUnitalRingHom.coe_one theorem mul_def (f g : α →ₙ+* α) : f * g = f.comp g := rfl #align non_unital_ring_hom.mul_def NonUnitalRingHom.mul_def @[simp] theorem coe_mul (f g : α →ₙ+* α) : ⇑(f * g) = f ∘ g := rfl #align non_unital_ring_hom.coe_mul NonUnitalRingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →ₙ+* γ} {f : α →ₙ+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_right NonUnitalRingHom.cancel_right @[simp] theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_left NonUnitalRingHom.cancel_left end NonUnitalRingHom /-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too. This extends from both `MonoidHom` and `MonoidWithZeroHom` in order to put the fields in a sensible order, even though `MonoidWithZeroHom` already extends `MonoidHom`. -/ structure RingHom (α : Type*) (β : Type*) [NonAssocSemiring α] [NonAssocSemiring β] extends α →* β, α →+ β, α →ₙ+* β, α →*₀ β #align ring_hom RingHom /-- `α →+* β` denotes the type of ring homomorphisms from `α` to `β`. -/ infixr:25 " →+* " => RingHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid with zero homomorphism `α →*₀ β`. The `simp`-normal form is `(f : α →*₀ β)`. -/ add_decl_doc RingHom.toMonoidWithZeroHom #align ring_hom.to_monoid_with_zero_hom RingHom.toMonoidWithZeroHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid homomorphism `α →* β`. The `simp`-normal form is `(f : α →* β)`. -/ add_decl_doc RingHom.toMonoidHom #align ring_hom.to_monoid_hom RingHom.toMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc RingHom.toAddMonoidHom #align ring_hom.to_add_monoid_hom RingHom.toAddMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a non-unital ring homomorphism `α →ₙ+* β`. The `simp`-normal form is `(f : α →ₙ+* β)`. -/ add_decl_doc RingHom.toNonUnitalRingHom #align ring_hom.to_non_unital_ring_hom RingHom.toNonUnitalRingHom section RingHomClass /-- `RingHomClass F α β` states that `F` is a type of (semi)ring homomorphisms. You should extend this class when you extend `RingHom`. This extends from both `MonoidHomClass` and `MonoidWithZeroHomClass` in order to put the fields in a sensible order, even though `MonoidWithZeroHomClass` already extends `MonoidHomClass`. -/ class RingHomClass (F : Type*) (α β : outParam (Type*)) [NonAssocSemiring α] [NonAssocSemiring β] extends MonoidHomClass F α β, AddMonoidHomClass F α β, MonoidWithZeroHomClass F α β #align ring_hom_class RingHomClass set_option linter.deprecated false in /-- Ring homomorphisms preserve `bit1`. -/ @[simp] lemma map_bit1 [NonAssocSemiring α] [NonAssocSemiring β] [RingHomClass F α β] (f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) := by simp [bit1] #align map_bit1 map_bit1 -- Porting note: marked `{}` rather than `[]` to prevent dangerous instances variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} [RingHomClass F α β] /-- Turn an element of a type `F` satisfying `RingHomClass F α β` into an actual `RingHom`. This is declared as the default coercion from `F` to `α →+* β`. -/ @[coe] def RingHomClass.toRingHom (f : F) : α →+* β := { (f : α →* β), (f : α →+ β) with } /-- Any type satisfying `RingHomClass` can be cast into `RingHom` via `RingHomClass.toRingHom`. -/ instance : CoeTC F (α →+* β) := ⟨RingHomClass.toRingHom⟩ instance (priority := 100) RingHomClass.toNonUnitalRingHomClass : NonUnitalRingHomClass F α β := { ‹RingHomClass F α β› with } #align ring_hom_class.to_non_unital_ring_hom_class RingHomClass.toNonUnitalRingHomClass end RingHomClass namespace RingHom section coe /-! Throughout this section, some `Semiring` arguments are specified with `{}` instead of `[]`. See note [implicit instance arguments]. -/ variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} instance instRingHomClass : RingHomClass (α →+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := RingHom.map_add' map_zero := RingHom.map_zero' map_mul f := f.map_mul' map_one f := f.map_one' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -- -/ -- instance : CoeFun (α →+* β) fun _ => α → β := -- ⟨RingHom.toFun⟩ initialize_simps_projections RingHom (toFun → apply) -- Porting note: is this lemma still needed in Lean4? -- Porting note: because `f.toFun` really means `f.toMonoidHom.toOneHom.toFun` and -- `toMonoidHom_eq_coe` wants to simplify `f.toMonoidHom` to `(↑f : M →* N)`, this can't -- be a simp lemma anymore -- @[simp] theorem toFun_eq_coe (f : α →+* β) : f.toFun = f := rfl #align ring_hom.to_fun_eq_coe RingHom.toFun_eq_coe @[simp] theorem coe_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α → β) = f := rfl #align ring_hom.coe_mk RingHom.coe_mk @[simp] theorem coe_coe {F : Type*} [RingHomClass F α β] (f : F) : ((f : α →+* β) : α → β) = f := rfl #align ring_hom.coe_coe RingHom.coe_coe attribute [coe] RingHom.toMonoidHom instance coeToMonoidHom : Coe (α →+* β) (α →* β) := ⟨RingHom.toMonoidHom⟩ #align ring_hom.has_coe_monoid_hom RingHom.coeToMonoidHom -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_monoid_hom @[simp] theorem toMonoidHom_eq_coe (f : α →+* β) : f.toMonoidHom = f := rfl #align ring_hom.to_monoid_hom_eq_coe RingHom.toMonoidHom_eq_coe -- Porting note: this can't be a simp lemma anymore -- @[simp] theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f := by rfl #align ring_hom.to_monoid_with_zero_hom_eq_coe RingHom.toMonoidWithZeroHom_eq_coe @[simp] theorem coe_monoidHom_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α →* β) = f := rfl #align ring_hom.coe_monoid_hom_mk RingHom.coe_monoidHom_mk -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_add_monoid_hom @[simp] theorem toAddMonoidHom_eq_coe (f : α →+* β) : f.toAddMonoidHom = f := rfl #align ring_hom.to_add_monoid_hom_eq_coe RingHom.toAddMonoidHom_eq_coe @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃ h₄) : ((⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨⟨f, h₃⟩, h₄⟩ := rfl #align ring_hom.coe_add_monoid_hom_mk RingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ def copy (f : α →+* β) (f' : α → β) (h : f' = f) : α →+* β := { f.toMonoidWithZeroHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align ring_hom.copy RingHom.copy @[simp] theorem coe_copy (f : α →+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align ring_hom.coe_copy RingHom.coe_copy theorem copy_eq (f : α →+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align ring_hom.copy_eq RingHom.copy_eq end coe section variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} (f : α →+* β) {x y : α} theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x := FunLike.congr_fun h x #align ring_hom.congr_fun RingHom.congr_fun theorem congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y := FunLike.congr_arg f h #align ring_hom.congr_arg RingHom.congr_arg theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g := FunLike.coe_injective h #align ring_hom.coe_inj RingHom.coe_inj @[ext] theorem ext ⦃f g : α →+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align ring_hom.ext RingHom.ext theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align ring_hom.ext_iff RingHom.ext_iff @[simp] theorem mk_coe (f : α →+* β) (h₁ h₂ h₃ h₄) : RingHom.mk ⟨⟨f, h₁⟩, h₂⟩ h₃ h₄ = f := ext fun _ => rfl #align ring_hom.mk_coe RingHom.mk_coe theorem coe_addMonoidHom_injective : Injective (fun f : α →+* β => (f : α →+ β)) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align ring_hom.coe_add_monoid_hom_injective RingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_monoidHom_injective : Injective (fun f : α →+* β => (f : α →* β)) := fun _ _ h => ext <| MonoidHom.congr_fun h #align ring_hom.coe_monoid_hom_injective RingHom.coe_monoidHom_injective /-- Ring homomorphisms map zero to zero. -/ protected theorem map_zero (f : α →+* β) : f 0 = 0 := map_zero f #align ring_hom.map_zero RingHom.map_zero /-- Ring homomorphisms map one to one. -/ protected theorem map_one (f : α →+* β) : f 1 = 1 := map_one f #align ring_hom.map_one RingHom.map_one /-- Ring homomorphisms preserve addition. -/ protected theorem map_add (f : α →+* β) : ∀ a b, f (a + b) = f a + f b := map_add f #align ring_hom.map_add RingHom.map_add /-- Ring homomorphisms preserve multiplication. -/ protected theorem map_mul (f : α →+* β) : ∀ a b, f (a * b) = f a * f b := map_mul f #align ring_hom.map_mul RingHom.map_mul @[simp] theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 0 1) = ite p 0 1 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_zero_one RingHom.map_ite_zero_one @[simp] theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 1 0) = ite p 1 0 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_one_zero RingHom.map_ite_one_zero /-- `f : α →+* β` has a trivial codomain iff `f 1 = 0`. -/ theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 := by rw [map_one, eq_comm] #align ring_hom.codomain_trivial_iff_map_one_eq_zero RingHom.codomain_trivial_iff_map_one_eq_zero /-- `f : α →+* β` has a trivial codomain iff it has a trivial range. -/ theorem codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 := f.codomain_trivial_iff_map_one_eq_zero.trans ⟨fun h x => by rw [← mul_one x, map_mul, h, mul_zero], fun h => h 1⟩ #align ring_hom.codomain_trivial_iff_range_trivial RingHom.codomain_trivial_iff_range_trivial /-- `f : α →+* β` doesn't map `1` to `0` if `β` is nontrivial -/ theorem map_one_ne_zero [Nontrivial β] : f 1 ≠ 0 := mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one #align ring_hom.map_one_ne_zero RingHom.map_one_ne_zero /-- If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. -/ theorem domain_nontrivial [Nontrivial β] : Nontrivial α := ⟨⟨1, 0, mt (fun h => show f 1 = 0 by rw [h, map_zero]) f.map_one_ne_zero⟩⟩ #align ring_hom.domain_nontrivial RingHom.domain_nontrivial theorem codomain_trivial (f : α →+* β) [h : Subsingleton α] : Subsingleton β := (subsingleton_or_nontrivial β).resolve_right fun _ => not_nontrivial_iff_subsingleton.mpr h f.domain_nontrivial #align ring_hom.codomain_trivial RingHom.codomain_trivial end /-- Ring homomorphisms preserve additive inverse. -/ protected theorem map_neg [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x : α) : f (-x) = -f x := map_neg f x #align ring_hom.map_neg RingHom.map_neg /-- Ring homomorphisms preserve subtraction. -/ protected theorem map_sub [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x y : α) : f (x - y) = f x - f y := map_sub f x y #align ring_hom.map_sub RingHom.map_sub /-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/ def mk' [NonAssocSemiring α] [NonAssocRing β] (f : α →* β) (map_add : ∀ a b, f (a + b) = f a + f b) : α →+* β := { AddMonoidHom.mk' f map_add, f with } #align ring_hom.mk' RingHom.mk' variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} /-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;> intros <;>

rfl

/-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;> intros <;>

Mathlib.Algebra.Ring.Hom.Defs.631_0.KyHvVYrIs9pW9ZQ

/-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α

Mathlib_Algebra_Ring_Hom_Defs

case refine'_3 F : Type u_1 α✝ : Type u_2 β : Type u_3 γ : Type u_4 x✝¹ : NonAssocSemiring α✝ x✝ : NonAssocSemiring β α : Type u_5 inst✝ : NonAssocSemiring α ⊢ OneHom.toFun (↑{ toOneHom := { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) }, map_mul' := (_ : ∀ (x y : α), OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } (x * y) = OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } (x * y)) }) 0 = 0

/- Copyright (c) 2019 Amelia Livingston. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Amelia Livingston, Jireh Loreaux -/ import Mathlib.Algebra.Ring.Defs import Mathlib.Algebra.Ring.Basic import Mathlib.Data.Pi.Algebra #align_import algebra.hom.ring from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9" /-! # Homomorphisms of semirings and rings This file defines bundled homomorphisms of (non-unital) semirings and rings. As with monoid and groups, we use the same structure `RingHom a β`, a.k.a. `α →+* β`, for both types of homomorphisms. ## Main definitions * `NonUnitalRingHom`: Non-unital (semi)ring homomorphisms. Additive monoid homomorphism which preserve multiplication. * `RingHom`: (Semi)ring homomorphisms. Monoid homomorphisms which are also additive monoid homomorphism. ## Notations * `→ₙ+*`: Non-unital (semi)ring homs * `→+*`: (Semi)ring homs ## Implementation notes * There's a coercion from bundled homs to fun, and the canonical notation is to use the bundled hom as a function via this coercion. * There is no `SemiringHom` -- the idea is that `RingHom` is used. The constructor for a `RingHom` between semirings needs a proof of `map_zero`, `map_one` and `map_add` as well as `map_mul`; a separate constructor `RingHom.mk'` will construct ring homs between rings from monoid homs given only a proof that addition is preserved. ## Tags `RingHom`, `SemiringHom` -/ open Function variable {F α β γ : Type*} /-- Bundled non-unital semiring homomorphisms `α →ₙ+* β`; use this for bundled non-unital ring homomorphisms too. When possible, instead of parametrizing results over `(f : α →ₙ+* β)`, you should parametrize over `(F : Type*) [NonUnitalRingHomClass F α β] (f : F)`. When you extend this structure, make sure to extend `NonUnitalRingHomClass`. -/ structure NonUnitalRingHom (α β : Type*) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends α →ₙ* β, α →+ β #align non_unital_ring_hom NonUnitalRingHom /-- `α →ₙ+* β` denotes the type of non-unital ring homomorphisms from `α` to `β`. -/ infixr:25 " →ₙ+* " => NonUnitalRingHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as a semigroup homomorphism `α →ₙ* β`. The `simp`-normal form is `(f : α →ₙ* β)`. -/ add_decl_doc NonUnitalRingHom.toMulHom #align non_unital_ring_hom.to_mul_hom NonUnitalRingHom.toMulHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc NonUnitalRingHom.toAddMonoidHom #align non_unital_ring_hom.to_add_monoid_hom NonUnitalRingHom.toAddMonoidHom section NonUnitalRingHomClass /-- `NonUnitalRingHomClass F α β` states that `F` is a type of non-unital (semi)ring homomorphisms. You should extend this class when you extend `NonUnitalRingHom`. -/ class NonUnitalRingHomClass (F : Type*) (α β : outParam (Type*)) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends MulHomClass F α β, AddMonoidHomClass F α β #align non_unital_ring_hom_class NonUnitalRingHomClass variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] [NonUnitalRingHomClass F α β] /-- Turn an element of a type `F` satisfying `NonUnitalRingHomClass F α β` into an actual `NonUnitalRingHom`. This is declared as the default coercion from `F` to `α →ₙ+* β`. -/ @[coe] def NonUnitalRingHomClass.toNonUnitalRingHom (f : F) : α →ₙ+* β := { (f : α →ₙ* β), (f : α →+ β) with } /-- Any type satisfying `NonUnitalRingHomClass` can be cast into `NonUnitalRingHom` via `NonUnitalRingHomClass.toNonUnitalRingHom`. -/ instance : CoeTC F (α →ₙ+* β) := ⟨NonUnitalRingHomClass.toNonUnitalRingHom⟩ end NonUnitalRingHomClass namespace NonUnitalRingHom section coe variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] instance : NonUnitalRingHomClass (α →ₙ+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := NonUnitalRingHom.map_add' map_zero := NonUnitalRingHom.map_zero' map_mul f := f.map_mul' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -/ -- instance : CoeFun (α →ₙ+* β) fun _ => α → β := -- ⟨fun f => f.toFun⟩ -- Porting note: removed due to new `coe` in Lean4 #noalign non_unital_ring_hom.to_fun_eq_coe #noalign non_unital_ring_hom.coe_mk #noalign non_unital_ring_hom.coe_coe initialize_simps_projections NonUnitalRingHom (toFun → apply) @[simp] theorem coe_toMulHom (f : α →ₙ+* β) : ⇑f.toMulHom = f := rfl #align non_unital_ring_hom.coe_to_mul_hom NonUnitalRingHom.coe_toMulHom @[simp] theorem coe_mulHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →ₙ* β) = ⟨f, h₁⟩ := rfl #align non_unital_ring_hom.coe_mul_hom_mk NonUnitalRingHom.coe_mulHom_mk theorem coe_toAddMonoidHom (f : α →ₙ+* β) : ⇑f.toAddMonoidHom = f := rfl #align non_unital_ring_hom.coe_to_add_monoid_hom NonUnitalRingHom.coe_toAddMonoidHom @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →+ β) = ⟨⟨f, h₂⟩, h₃⟩ := rfl #align non_unital_ring_hom.coe_add_monoid_hom_mk NonUnitalRingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : α →ₙ+* β := { f.toMulHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align non_unital_ring_hom.copy NonUnitalRingHom.copy @[simp] theorem coe_copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align non_unital_ring_hom.coe_copy NonUnitalRingHom.coe_copy theorem copy_eq (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align non_unital_ring_hom.copy_eq NonUnitalRingHom.copy_eq end coe section variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] variable (f : α →ₙ+* β) {x y : α} @[ext] theorem ext ⦃f g : α →ₙ+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align non_unital_ring_hom.ext NonUnitalRingHom.ext theorem ext_iff {f g : α →ₙ+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align non_unital_ring_hom.ext_iff NonUnitalRingHom.ext_iff @[simp] theorem mk_coe (f : α →ₙ+* β) (h₁ h₂ h₃) : NonUnitalRingHom.mk (MulHom.mk f h₁) h₂ h₃ = f := ext fun _ => rfl #align non_unital_ring_hom.mk_coe NonUnitalRingHom.mk_coe theorem coe_addMonoidHom_injective : Injective fun f : α →ₙ+* β => (f : α →+ β) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align non_unital_ring_hom.coe_add_monoid_hom_injective NonUnitalRingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_mulHom_injective : Injective fun f : α →ₙ+* β => (f : α →ₙ* β) := fun _ _ h => ext <| MulHom.congr_fun h #align non_unital_ring_hom.coe_mul_hom_injective NonUnitalRingHom.coe_mulHom_injective end variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/ protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by refine' { toFun := id.. } <;> intros <;> rfl #align non_unital_ring_hom.id NonUnitalRingHom.id instance : Zero (α →ₙ+* β) := ⟨{ toFun := 0, map_mul' := fun _ _ => (mul_zero (0 : β)).symm, map_zero' := rfl, map_add' := fun _ _ => (add_zero (0 : β)).symm }⟩ instance : Inhabited (α →ₙ+* β) := ⟨0⟩ @[simp] theorem coe_zero : ⇑(0 : α →ₙ+* β) = 0 := rfl #align non_unital_ring_hom.coe_zero NonUnitalRingHom.coe_zero @[simp] theorem zero_apply (x : α) : (0 : α →ₙ+* β) x = 0 := rfl #align non_unital_ring_hom.zero_apply NonUnitalRingHom.zero_apply @[simp] theorem id_apply (x : α) : NonUnitalRingHom.id α x = x := rfl #align non_unital_ring_hom.id_apply NonUnitalRingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (NonUnitalRingHom.id α : α →+ α) = AddMonoidHom.id α := rfl #align non_unital_ring_hom.coe_add_monoid_hom_id NonUnitalRingHom.coe_addMonoidHom_id @[simp] theorem coe_mulHom_id : (NonUnitalRingHom.id α : α →ₙ* α) = MulHom.id α := rfl #align non_unital_ring_hom.coe_mul_hom_id NonUnitalRingHom.coe_mulHom_id variable [NonUnitalNonAssocSemiring γ] /-- Composition of non-unital ring homomorphisms is a non-unital ring homomorphism. -/ def comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : α →ₙ+* γ := { g.toMulHom.comp f.toMulHom, g.toAddMonoidHom.comp f.toAddMonoidHom with } #align non_unital_ring_hom.comp NonUnitalRingHom.comp /-- Composition of non-unital ring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonUnitalNonAssocSemiring δ} (f : α →ₙ+* β) (g : β →ₙ+* γ) (h : γ →ₙ+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align non_unital_ring_hom.comp_assoc NonUnitalRingHom.comp_assoc @[simp] theorem coe_comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : ⇑(g.comp f) = g ∘ f := rfl #align non_unital_ring_hom.coe_comp NonUnitalRingHom.coe_comp @[simp] theorem comp_apply (g : β →ₙ+* γ) (f : α →ₙ+* β) (x : α) : g.comp f x = g (f x) := rfl #align non_unital_ring_hom.comp_apply NonUnitalRingHom.comp_apply variable (g : β →ₙ+* γ) (f : α →ₙ+* β) @[simp] theorem coe_comp_addMonoidHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : AddMonoidHom.mk ⟨g ∘ f, (g.comp f).map_zero'⟩ (g.comp f).map_add' = (g : β →+ γ).comp f := rfl #align non_unital_ring_hom.coe_comp_add_monoid_hom NonUnitalRingHom.coe_comp_addMonoidHom @[simp] theorem coe_comp_mulHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : MulHom.mk (g ∘ f) (g.comp f).map_mul' = (g : β →ₙ* γ).comp f := rfl #align non_unital_ring_hom.coe_comp_mul_hom NonUnitalRingHom.coe_comp_mulHom @[simp] theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 := by ext simp #align non_unital_ring_hom.comp_zero NonUnitalRingHom.comp_zero @[simp] theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 := by ext rfl #align non_unital_ring_hom.zero_comp NonUnitalRingHom.zero_comp @[simp] theorem comp_id (f : α →ₙ+* β) : f.comp (NonUnitalRingHom.id α) = f := ext fun _ => rfl #align non_unital_ring_hom.comp_id NonUnitalRingHom.comp_id @[simp] theorem id_comp (f : α →ₙ+* β) : (NonUnitalRingHom.id β).comp f = f := ext fun _ => rfl #align non_unital_ring_hom.id_comp NonUnitalRingHom.id_comp instance : MonoidWithZero (α →ₙ+* α) where one := NonUnitalRingHom.id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ zero := 0 mul_zero := comp_zero zero_mul := zero_comp theorem one_def : (1 : α →ₙ+* α) = NonUnitalRingHom.id α := rfl #align non_unital_ring_hom.one_def NonUnitalRingHom.one_def @[simp] theorem coe_one : ⇑(1 : α →ₙ+* α) = id := rfl #align non_unital_ring_hom.coe_one NonUnitalRingHom.coe_one theorem mul_def (f g : α →ₙ+* α) : f * g = f.comp g := rfl #align non_unital_ring_hom.mul_def NonUnitalRingHom.mul_def @[simp] theorem coe_mul (f g : α →ₙ+* α) : ⇑(f * g) = f ∘ g := rfl #align non_unital_ring_hom.coe_mul NonUnitalRingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →ₙ+* γ} {f : α →ₙ+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_right NonUnitalRingHom.cancel_right @[simp] theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_left NonUnitalRingHom.cancel_left end NonUnitalRingHom /-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too. This extends from both `MonoidHom` and `MonoidWithZeroHom` in order to put the fields in a sensible order, even though `MonoidWithZeroHom` already extends `MonoidHom`. -/ structure RingHom (α : Type*) (β : Type*) [NonAssocSemiring α] [NonAssocSemiring β] extends α →* β, α →+ β, α →ₙ+* β, α →*₀ β #align ring_hom RingHom /-- `α →+* β` denotes the type of ring homomorphisms from `α` to `β`. -/ infixr:25 " →+* " => RingHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid with zero homomorphism `α →*₀ β`. The `simp`-normal form is `(f : α →*₀ β)`. -/ add_decl_doc RingHom.toMonoidWithZeroHom #align ring_hom.to_monoid_with_zero_hom RingHom.toMonoidWithZeroHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid homomorphism `α →* β`. The `simp`-normal form is `(f : α →* β)`. -/ add_decl_doc RingHom.toMonoidHom #align ring_hom.to_monoid_hom RingHom.toMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc RingHom.toAddMonoidHom #align ring_hom.to_add_monoid_hom RingHom.toAddMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a non-unital ring homomorphism `α →ₙ+* β`. The `simp`-normal form is `(f : α →ₙ+* β)`. -/ add_decl_doc RingHom.toNonUnitalRingHom #align ring_hom.to_non_unital_ring_hom RingHom.toNonUnitalRingHom section RingHomClass /-- `RingHomClass F α β` states that `F` is a type of (semi)ring homomorphisms. You should extend this class when you extend `RingHom`. This extends from both `MonoidHomClass` and `MonoidWithZeroHomClass` in order to put the fields in a sensible order, even though `MonoidWithZeroHomClass` already extends `MonoidHomClass`. -/ class RingHomClass (F : Type*) (α β : outParam (Type*)) [NonAssocSemiring α] [NonAssocSemiring β] extends MonoidHomClass F α β, AddMonoidHomClass F α β, MonoidWithZeroHomClass F α β #align ring_hom_class RingHomClass set_option linter.deprecated false in /-- Ring homomorphisms preserve `bit1`. -/ @[simp] lemma map_bit1 [NonAssocSemiring α] [NonAssocSemiring β] [RingHomClass F α β] (f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) := by simp [bit1] #align map_bit1 map_bit1 -- Porting note: marked `{}` rather than `[]` to prevent dangerous instances variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} [RingHomClass F α β] /-- Turn an element of a type `F` satisfying `RingHomClass F α β` into an actual `RingHom`. This is declared as the default coercion from `F` to `α →+* β`. -/ @[coe] def RingHomClass.toRingHom (f : F) : α →+* β := { (f : α →* β), (f : α →+ β) with } /-- Any type satisfying `RingHomClass` can be cast into `RingHom` via `RingHomClass.toRingHom`. -/ instance : CoeTC F (α →+* β) := ⟨RingHomClass.toRingHom⟩ instance (priority := 100) RingHomClass.toNonUnitalRingHomClass : NonUnitalRingHomClass F α β := { ‹RingHomClass F α β› with } #align ring_hom_class.to_non_unital_ring_hom_class RingHomClass.toNonUnitalRingHomClass end RingHomClass namespace RingHom section coe /-! Throughout this section, some `Semiring` arguments are specified with `{}` instead of `[]`. See note [implicit instance arguments]. -/ variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} instance instRingHomClass : RingHomClass (α →+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := RingHom.map_add' map_zero := RingHom.map_zero' map_mul f := f.map_mul' map_one f := f.map_one' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -- -/ -- instance : CoeFun (α →+* β) fun _ => α → β := -- ⟨RingHom.toFun⟩ initialize_simps_projections RingHom (toFun → apply) -- Porting note: is this lemma still needed in Lean4? -- Porting note: because `f.toFun` really means `f.toMonoidHom.toOneHom.toFun` and -- `toMonoidHom_eq_coe` wants to simplify `f.toMonoidHom` to `(↑f : M →* N)`, this can't -- be a simp lemma anymore -- @[simp] theorem toFun_eq_coe (f : α →+* β) : f.toFun = f := rfl #align ring_hom.to_fun_eq_coe RingHom.toFun_eq_coe @[simp] theorem coe_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α → β) = f := rfl #align ring_hom.coe_mk RingHom.coe_mk @[simp] theorem coe_coe {F : Type*} [RingHomClass F α β] (f : F) : ((f : α →+* β) : α → β) = f := rfl #align ring_hom.coe_coe RingHom.coe_coe attribute [coe] RingHom.toMonoidHom instance coeToMonoidHom : Coe (α →+* β) (α →* β) := ⟨RingHom.toMonoidHom⟩ #align ring_hom.has_coe_monoid_hom RingHom.coeToMonoidHom -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_monoid_hom @[simp] theorem toMonoidHom_eq_coe (f : α →+* β) : f.toMonoidHom = f := rfl #align ring_hom.to_monoid_hom_eq_coe RingHom.toMonoidHom_eq_coe -- Porting note: this can't be a simp lemma anymore -- @[simp] theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f := by rfl #align ring_hom.to_monoid_with_zero_hom_eq_coe RingHom.toMonoidWithZeroHom_eq_coe @[simp] theorem coe_monoidHom_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α →* β) = f := rfl #align ring_hom.coe_monoid_hom_mk RingHom.coe_monoidHom_mk -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_add_monoid_hom @[simp] theorem toAddMonoidHom_eq_coe (f : α →+* β) : f.toAddMonoidHom = f := rfl #align ring_hom.to_add_monoid_hom_eq_coe RingHom.toAddMonoidHom_eq_coe @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃ h₄) : ((⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨⟨f, h₃⟩, h₄⟩ := rfl #align ring_hom.coe_add_monoid_hom_mk RingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ def copy (f : α →+* β) (f' : α → β) (h : f' = f) : α →+* β := { f.toMonoidWithZeroHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align ring_hom.copy RingHom.copy @[simp] theorem coe_copy (f : α →+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align ring_hom.coe_copy RingHom.coe_copy theorem copy_eq (f : α →+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align ring_hom.copy_eq RingHom.copy_eq end coe section variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} (f : α →+* β) {x y : α} theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x := FunLike.congr_fun h x #align ring_hom.congr_fun RingHom.congr_fun theorem congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y := FunLike.congr_arg f h #align ring_hom.congr_arg RingHom.congr_arg theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g := FunLike.coe_injective h #align ring_hom.coe_inj RingHom.coe_inj @[ext] theorem ext ⦃f g : α →+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align ring_hom.ext RingHom.ext theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align ring_hom.ext_iff RingHom.ext_iff @[simp] theorem mk_coe (f : α →+* β) (h₁ h₂ h₃ h₄) : RingHom.mk ⟨⟨f, h₁⟩, h₂⟩ h₃ h₄ = f := ext fun _ => rfl #align ring_hom.mk_coe RingHom.mk_coe theorem coe_addMonoidHom_injective : Injective (fun f : α →+* β => (f : α →+ β)) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align ring_hom.coe_add_monoid_hom_injective RingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_monoidHom_injective : Injective (fun f : α →+* β => (f : α →* β)) := fun _ _ h => ext <| MonoidHom.congr_fun h #align ring_hom.coe_monoid_hom_injective RingHom.coe_monoidHom_injective /-- Ring homomorphisms map zero to zero. -/ protected theorem map_zero (f : α →+* β) : f 0 = 0 := map_zero f #align ring_hom.map_zero RingHom.map_zero /-- Ring homomorphisms map one to one. -/ protected theorem map_one (f : α →+* β) : f 1 = 1 := map_one f #align ring_hom.map_one RingHom.map_one /-- Ring homomorphisms preserve addition. -/ protected theorem map_add (f : α →+* β) : ∀ a b, f (a + b) = f a + f b := map_add f #align ring_hom.map_add RingHom.map_add /-- Ring homomorphisms preserve multiplication. -/ protected theorem map_mul (f : α →+* β) : ∀ a b, f (a * b) = f a * f b := map_mul f #align ring_hom.map_mul RingHom.map_mul @[simp] theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 0 1) = ite p 0 1 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_zero_one RingHom.map_ite_zero_one @[simp] theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 1 0) = ite p 1 0 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_one_zero RingHom.map_ite_one_zero /-- `f : α →+* β` has a trivial codomain iff `f 1 = 0`. -/ theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 := by rw [map_one, eq_comm] #align ring_hom.codomain_trivial_iff_map_one_eq_zero RingHom.codomain_trivial_iff_map_one_eq_zero /-- `f : α →+* β` has a trivial codomain iff it has a trivial range. -/ theorem codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 := f.codomain_trivial_iff_map_one_eq_zero.trans ⟨fun h x => by rw [← mul_one x, map_mul, h, mul_zero], fun h => h 1⟩ #align ring_hom.codomain_trivial_iff_range_trivial RingHom.codomain_trivial_iff_range_trivial /-- `f : α →+* β` doesn't map `1` to `0` if `β` is nontrivial -/ theorem map_one_ne_zero [Nontrivial β] : f 1 ≠ 0 := mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one #align ring_hom.map_one_ne_zero RingHom.map_one_ne_zero /-- If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. -/ theorem domain_nontrivial [Nontrivial β] : Nontrivial α := ⟨⟨1, 0, mt (fun h => show f 1 = 0 by rw [h, map_zero]) f.map_one_ne_zero⟩⟩ #align ring_hom.domain_nontrivial RingHom.domain_nontrivial theorem codomain_trivial (f : α →+* β) [h : Subsingleton α] : Subsingleton β := (subsingleton_or_nontrivial β).resolve_right fun _ => not_nontrivial_iff_subsingleton.mpr h f.domain_nontrivial #align ring_hom.codomain_trivial RingHom.codomain_trivial end /-- Ring homomorphisms preserve additive inverse. -/ protected theorem map_neg [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x : α) : f (-x) = -f x := map_neg f x #align ring_hom.map_neg RingHom.map_neg /-- Ring homomorphisms preserve subtraction. -/ protected theorem map_sub [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x y : α) : f (x - y) = f x - f y := map_sub f x y #align ring_hom.map_sub RingHom.map_sub /-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/ def mk' [NonAssocSemiring α] [NonAssocRing β] (f : α →* β) (map_add : ∀ a b, f (a + b) = f a + f b) : α →+* β := { AddMonoidHom.mk' f map_add, f with } #align ring_hom.mk' RingHom.mk' variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} /-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;> intros <;>

rfl

/-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;> intros <;>

Mathlib.Algebra.Ring.Hom.Defs.631_0.KyHvVYrIs9pW9ZQ

/-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α

Mathlib_Algebra_Ring_Hom_Defs

case refine'_4 F : Type u_1 α✝ : Type u_2 β : Type u_3 γ : Type u_4 x✝² : NonAssocSemiring α✝ x✝¹ : NonAssocSemiring β α : Type u_5 inst✝ : NonAssocSemiring α x✝ y✝ : α ⊢ OneHom.toFun (↑{ toOneHom := { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) }, map_mul' := (_ : ∀ (x y : α), OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } (x * y) = OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } (x * y)) }) (x✝ + y✝) = OneHom.toFun (↑{ toOneHom := { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) }, map_mul' := (_ : ∀ (x y : α), OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } (x * y) = OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } (x * y)) }) x✝ + OneHom.toFun (↑{ toOneHom := { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) }, map_mul' := (_ : ∀ (x y : α), OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } (x * y) = OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } (x * y)) }) y✝

/- Copyright (c) 2019 Amelia Livingston. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Amelia Livingston, Jireh Loreaux -/ import Mathlib.Algebra.Ring.Defs import Mathlib.Algebra.Ring.Basic import Mathlib.Data.Pi.Algebra #align_import algebra.hom.ring from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9" /-! # Homomorphisms of semirings and rings This file defines bundled homomorphisms of (non-unital) semirings and rings. As with monoid and groups, we use the same structure `RingHom a β`, a.k.a. `α →+* β`, for both types of homomorphisms. ## Main definitions * `NonUnitalRingHom`: Non-unital (semi)ring homomorphisms. Additive monoid homomorphism which preserve multiplication. * `RingHom`: (Semi)ring homomorphisms. Monoid homomorphisms which are also additive monoid homomorphism. ## Notations * `→ₙ+*`: Non-unital (semi)ring homs * `→+*`: (Semi)ring homs ## Implementation notes * There's a coercion from bundled homs to fun, and the canonical notation is to use the bundled hom as a function via this coercion. * There is no `SemiringHom` -- the idea is that `RingHom` is used. The constructor for a `RingHom` between semirings needs a proof of `map_zero`, `map_one` and `map_add` as well as `map_mul`; a separate constructor `RingHom.mk'` will construct ring homs between rings from monoid homs given only a proof that addition is preserved. ## Tags `RingHom`, `SemiringHom` -/ open Function variable {F α β γ : Type*} /-- Bundled non-unital semiring homomorphisms `α →ₙ+* β`; use this for bundled non-unital ring homomorphisms too. When possible, instead of parametrizing results over `(f : α →ₙ+* β)`, you should parametrize over `(F : Type*) [NonUnitalRingHomClass F α β] (f : F)`. When you extend this structure, make sure to extend `NonUnitalRingHomClass`. -/ structure NonUnitalRingHom (α β : Type*) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends α →ₙ* β, α →+ β #align non_unital_ring_hom NonUnitalRingHom /-- `α →ₙ+* β` denotes the type of non-unital ring homomorphisms from `α` to `β`. -/ infixr:25 " →ₙ+* " => NonUnitalRingHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as a semigroup homomorphism `α →ₙ* β`. The `simp`-normal form is `(f : α →ₙ* β)`. -/ add_decl_doc NonUnitalRingHom.toMulHom #align non_unital_ring_hom.to_mul_hom NonUnitalRingHom.toMulHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc NonUnitalRingHom.toAddMonoidHom #align non_unital_ring_hom.to_add_monoid_hom NonUnitalRingHom.toAddMonoidHom section NonUnitalRingHomClass /-- `NonUnitalRingHomClass F α β` states that `F` is a type of non-unital (semi)ring homomorphisms. You should extend this class when you extend `NonUnitalRingHom`. -/ class NonUnitalRingHomClass (F : Type*) (α β : outParam (Type*)) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends MulHomClass F α β, AddMonoidHomClass F α β #align non_unital_ring_hom_class NonUnitalRingHomClass variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] [NonUnitalRingHomClass F α β] /-- Turn an element of a type `F` satisfying `NonUnitalRingHomClass F α β` into an actual `NonUnitalRingHom`. This is declared as the default coercion from `F` to `α →ₙ+* β`. -/ @[coe] def NonUnitalRingHomClass.toNonUnitalRingHom (f : F) : α →ₙ+* β := { (f : α →ₙ* β), (f : α →+ β) with } /-- Any type satisfying `NonUnitalRingHomClass` can be cast into `NonUnitalRingHom` via `NonUnitalRingHomClass.toNonUnitalRingHom`. -/ instance : CoeTC F (α →ₙ+* β) := ⟨NonUnitalRingHomClass.toNonUnitalRingHom⟩ end NonUnitalRingHomClass namespace NonUnitalRingHom section coe variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] instance : NonUnitalRingHomClass (α →ₙ+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := NonUnitalRingHom.map_add' map_zero := NonUnitalRingHom.map_zero' map_mul f := f.map_mul' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -/ -- instance : CoeFun (α →ₙ+* β) fun _ => α → β := -- ⟨fun f => f.toFun⟩ -- Porting note: removed due to new `coe` in Lean4 #noalign non_unital_ring_hom.to_fun_eq_coe #noalign non_unital_ring_hom.coe_mk #noalign non_unital_ring_hom.coe_coe initialize_simps_projections NonUnitalRingHom (toFun → apply) @[simp] theorem coe_toMulHom (f : α →ₙ+* β) : ⇑f.toMulHom = f := rfl #align non_unital_ring_hom.coe_to_mul_hom NonUnitalRingHom.coe_toMulHom @[simp] theorem coe_mulHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →ₙ* β) = ⟨f, h₁⟩ := rfl #align non_unital_ring_hom.coe_mul_hom_mk NonUnitalRingHom.coe_mulHom_mk theorem coe_toAddMonoidHom (f : α →ₙ+* β) : ⇑f.toAddMonoidHom = f := rfl #align non_unital_ring_hom.coe_to_add_monoid_hom NonUnitalRingHom.coe_toAddMonoidHom @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →+ β) = ⟨⟨f, h₂⟩, h₃⟩ := rfl #align non_unital_ring_hom.coe_add_monoid_hom_mk NonUnitalRingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : α →ₙ+* β := { f.toMulHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align non_unital_ring_hom.copy NonUnitalRingHom.copy @[simp] theorem coe_copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align non_unital_ring_hom.coe_copy NonUnitalRingHom.coe_copy theorem copy_eq (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align non_unital_ring_hom.copy_eq NonUnitalRingHom.copy_eq end coe section variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] variable (f : α →ₙ+* β) {x y : α} @[ext] theorem ext ⦃f g : α →ₙ+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align non_unital_ring_hom.ext NonUnitalRingHom.ext theorem ext_iff {f g : α →ₙ+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align non_unital_ring_hom.ext_iff NonUnitalRingHom.ext_iff @[simp] theorem mk_coe (f : α →ₙ+* β) (h₁ h₂ h₃) : NonUnitalRingHom.mk (MulHom.mk f h₁) h₂ h₃ = f := ext fun _ => rfl #align non_unital_ring_hom.mk_coe NonUnitalRingHom.mk_coe theorem coe_addMonoidHom_injective : Injective fun f : α →ₙ+* β => (f : α →+ β) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align non_unital_ring_hom.coe_add_monoid_hom_injective NonUnitalRingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_mulHom_injective : Injective fun f : α →ₙ+* β => (f : α →ₙ* β) := fun _ _ h => ext <| MulHom.congr_fun h #align non_unital_ring_hom.coe_mul_hom_injective NonUnitalRingHom.coe_mulHom_injective end variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/ protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by refine' { toFun := id.. } <;> intros <;> rfl #align non_unital_ring_hom.id NonUnitalRingHom.id instance : Zero (α →ₙ+* β) := ⟨{ toFun := 0, map_mul' := fun _ _ => (mul_zero (0 : β)).symm, map_zero' := rfl, map_add' := fun _ _ => (add_zero (0 : β)).symm }⟩ instance : Inhabited (α →ₙ+* β) := ⟨0⟩ @[simp] theorem coe_zero : ⇑(0 : α →ₙ+* β) = 0 := rfl #align non_unital_ring_hom.coe_zero NonUnitalRingHom.coe_zero @[simp] theorem zero_apply (x : α) : (0 : α →ₙ+* β) x = 0 := rfl #align non_unital_ring_hom.zero_apply NonUnitalRingHom.zero_apply @[simp] theorem id_apply (x : α) : NonUnitalRingHom.id α x = x := rfl #align non_unital_ring_hom.id_apply NonUnitalRingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (NonUnitalRingHom.id α : α →+ α) = AddMonoidHom.id α := rfl #align non_unital_ring_hom.coe_add_monoid_hom_id NonUnitalRingHom.coe_addMonoidHom_id @[simp] theorem coe_mulHom_id : (NonUnitalRingHom.id α : α →ₙ* α) = MulHom.id α := rfl #align non_unital_ring_hom.coe_mul_hom_id NonUnitalRingHom.coe_mulHom_id variable [NonUnitalNonAssocSemiring γ] /-- Composition of non-unital ring homomorphisms is a non-unital ring homomorphism. -/ def comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : α →ₙ+* γ := { g.toMulHom.comp f.toMulHom, g.toAddMonoidHom.comp f.toAddMonoidHom with } #align non_unital_ring_hom.comp NonUnitalRingHom.comp /-- Composition of non-unital ring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonUnitalNonAssocSemiring δ} (f : α →ₙ+* β) (g : β →ₙ+* γ) (h : γ →ₙ+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align non_unital_ring_hom.comp_assoc NonUnitalRingHom.comp_assoc @[simp] theorem coe_comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : ⇑(g.comp f) = g ∘ f := rfl #align non_unital_ring_hom.coe_comp NonUnitalRingHom.coe_comp @[simp] theorem comp_apply (g : β →ₙ+* γ) (f : α →ₙ+* β) (x : α) : g.comp f x = g (f x) := rfl #align non_unital_ring_hom.comp_apply NonUnitalRingHom.comp_apply variable (g : β →ₙ+* γ) (f : α →ₙ+* β) @[simp] theorem coe_comp_addMonoidHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : AddMonoidHom.mk ⟨g ∘ f, (g.comp f).map_zero'⟩ (g.comp f).map_add' = (g : β →+ γ).comp f := rfl #align non_unital_ring_hom.coe_comp_add_monoid_hom NonUnitalRingHom.coe_comp_addMonoidHom @[simp] theorem coe_comp_mulHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : MulHom.mk (g ∘ f) (g.comp f).map_mul' = (g : β →ₙ* γ).comp f := rfl #align non_unital_ring_hom.coe_comp_mul_hom NonUnitalRingHom.coe_comp_mulHom @[simp] theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 := by ext simp #align non_unital_ring_hom.comp_zero NonUnitalRingHom.comp_zero @[simp] theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 := by ext rfl #align non_unital_ring_hom.zero_comp NonUnitalRingHom.zero_comp @[simp] theorem comp_id (f : α →ₙ+* β) : f.comp (NonUnitalRingHom.id α) = f := ext fun _ => rfl #align non_unital_ring_hom.comp_id NonUnitalRingHom.comp_id @[simp] theorem id_comp (f : α →ₙ+* β) : (NonUnitalRingHom.id β).comp f = f := ext fun _ => rfl #align non_unital_ring_hom.id_comp NonUnitalRingHom.id_comp instance : MonoidWithZero (α →ₙ+* α) where one := NonUnitalRingHom.id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ zero := 0 mul_zero := comp_zero zero_mul := zero_comp theorem one_def : (1 : α →ₙ+* α) = NonUnitalRingHom.id α := rfl #align non_unital_ring_hom.one_def NonUnitalRingHom.one_def @[simp] theorem coe_one : ⇑(1 : α →ₙ+* α) = id := rfl #align non_unital_ring_hom.coe_one NonUnitalRingHom.coe_one theorem mul_def (f g : α →ₙ+* α) : f * g = f.comp g := rfl #align non_unital_ring_hom.mul_def NonUnitalRingHom.mul_def @[simp] theorem coe_mul (f g : α →ₙ+* α) : ⇑(f * g) = f ∘ g := rfl #align non_unital_ring_hom.coe_mul NonUnitalRingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →ₙ+* γ} {f : α →ₙ+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_right NonUnitalRingHom.cancel_right @[simp] theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_left NonUnitalRingHom.cancel_left end NonUnitalRingHom /-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too. This extends from both `MonoidHom` and `MonoidWithZeroHom` in order to put the fields in a sensible order, even though `MonoidWithZeroHom` already extends `MonoidHom`. -/ structure RingHom (α : Type*) (β : Type*) [NonAssocSemiring α] [NonAssocSemiring β] extends α →* β, α →+ β, α →ₙ+* β, α →*₀ β #align ring_hom RingHom /-- `α →+* β` denotes the type of ring homomorphisms from `α` to `β`. -/ infixr:25 " →+* " => RingHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid with zero homomorphism `α →*₀ β`. The `simp`-normal form is `(f : α →*₀ β)`. -/ add_decl_doc RingHom.toMonoidWithZeroHom #align ring_hom.to_monoid_with_zero_hom RingHom.toMonoidWithZeroHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid homomorphism `α →* β`. The `simp`-normal form is `(f : α →* β)`. -/ add_decl_doc RingHom.toMonoidHom #align ring_hom.to_monoid_hom RingHom.toMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc RingHom.toAddMonoidHom #align ring_hom.to_add_monoid_hom RingHom.toAddMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a non-unital ring homomorphism `α →ₙ+* β`. The `simp`-normal form is `(f : α →ₙ+* β)`. -/ add_decl_doc RingHom.toNonUnitalRingHom #align ring_hom.to_non_unital_ring_hom RingHom.toNonUnitalRingHom section RingHomClass /-- `RingHomClass F α β` states that `F` is a type of (semi)ring homomorphisms. You should extend this class when you extend `RingHom`. This extends from both `MonoidHomClass` and `MonoidWithZeroHomClass` in order to put the fields in a sensible order, even though `MonoidWithZeroHomClass` already extends `MonoidHomClass`. -/ class RingHomClass (F : Type*) (α β : outParam (Type*)) [NonAssocSemiring α] [NonAssocSemiring β] extends MonoidHomClass F α β, AddMonoidHomClass F α β, MonoidWithZeroHomClass F α β #align ring_hom_class RingHomClass set_option linter.deprecated false in /-- Ring homomorphisms preserve `bit1`. -/ @[simp] lemma map_bit1 [NonAssocSemiring α] [NonAssocSemiring β] [RingHomClass F α β] (f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) := by simp [bit1] #align map_bit1 map_bit1 -- Porting note: marked `{}` rather than `[]` to prevent dangerous instances variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} [RingHomClass F α β] /-- Turn an element of a type `F` satisfying `RingHomClass F α β` into an actual `RingHom`. This is declared as the default coercion from `F` to `α →+* β`. -/ @[coe] def RingHomClass.toRingHom (f : F) : α →+* β := { (f : α →* β), (f : α →+ β) with } /-- Any type satisfying `RingHomClass` can be cast into `RingHom` via `RingHomClass.toRingHom`. -/ instance : CoeTC F (α →+* β) := ⟨RingHomClass.toRingHom⟩ instance (priority := 100) RingHomClass.toNonUnitalRingHomClass : NonUnitalRingHomClass F α β := { ‹RingHomClass F α β› with } #align ring_hom_class.to_non_unital_ring_hom_class RingHomClass.toNonUnitalRingHomClass end RingHomClass namespace RingHom section coe /-! Throughout this section, some `Semiring` arguments are specified with `{}` instead of `[]`. See note [implicit instance arguments]. -/ variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} instance instRingHomClass : RingHomClass (α →+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := RingHom.map_add' map_zero := RingHom.map_zero' map_mul f := f.map_mul' map_one f := f.map_one' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -- -/ -- instance : CoeFun (α →+* β) fun _ => α → β := -- ⟨RingHom.toFun⟩ initialize_simps_projections RingHom (toFun → apply) -- Porting note: is this lemma still needed in Lean4? -- Porting note: because `f.toFun` really means `f.toMonoidHom.toOneHom.toFun` and -- `toMonoidHom_eq_coe` wants to simplify `f.toMonoidHom` to `(↑f : M →* N)`, this can't -- be a simp lemma anymore -- @[simp] theorem toFun_eq_coe (f : α →+* β) : f.toFun = f := rfl #align ring_hom.to_fun_eq_coe RingHom.toFun_eq_coe @[simp] theorem coe_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α → β) = f := rfl #align ring_hom.coe_mk RingHom.coe_mk @[simp] theorem coe_coe {F : Type*} [RingHomClass F α β] (f : F) : ((f : α →+* β) : α → β) = f := rfl #align ring_hom.coe_coe RingHom.coe_coe attribute [coe] RingHom.toMonoidHom instance coeToMonoidHom : Coe (α →+* β) (α →* β) := ⟨RingHom.toMonoidHom⟩ #align ring_hom.has_coe_monoid_hom RingHom.coeToMonoidHom -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_monoid_hom @[simp] theorem toMonoidHom_eq_coe (f : α →+* β) : f.toMonoidHom = f := rfl #align ring_hom.to_monoid_hom_eq_coe RingHom.toMonoidHom_eq_coe -- Porting note: this can't be a simp lemma anymore -- @[simp] theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f := by rfl #align ring_hom.to_monoid_with_zero_hom_eq_coe RingHom.toMonoidWithZeroHom_eq_coe @[simp] theorem coe_monoidHom_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α →* β) = f := rfl #align ring_hom.coe_monoid_hom_mk RingHom.coe_monoidHom_mk -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_add_monoid_hom @[simp] theorem toAddMonoidHom_eq_coe (f : α →+* β) : f.toAddMonoidHom = f := rfl #align ring_hom.to_add_monoid_hom_eq_coe RingHom.toAddMonoidHom_eq_coe @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃ h₄) : ((⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨⟨f, h₃⟩, h₄⟩ := rfl #align ring_hom.coe_add_monoid_hom_mk RingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ def copy (f : α →+* β) (f' : α → β) (h : f' = f) : α →+* β := { f.toMonoidWithZeroHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align ring_hom.copy RingHom.copy @[simp] theorem coe_copy (f : α →+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align ring_hom.coe_copy RingHom.coe_copy theorem copy_eq (f : α →+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align ring_hom.copy_eq RingHom.copy_eq end coe section variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} (f : α →+* β) {x y : α} theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x := FunLike.congr_fun h x #align ring_hom.congr_fun RingHom.congr_fun theorem congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y := FunLike.congr_arg f h #align ring_hom.congr_arg RingHom.congr_arg theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g := FunLike.coe_injective h #align ring_hom.coe_inj RingHom.coe_inj @[ext] theorem ext ⦃f g : α →+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align ring_hom.ext RingHom.ext theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align ring_hom.ext_iff RingHom.ext_iff @[simp] theorem mk_coe (f : α →+* β) (h₁ h₂ h₃ h₄) : RingHom.mk ⟨⟨f, h₁⟩, h₂⟩ h₃ h₄ = f := ext fun _ => rfl #align ring_hom.mk_coe RingHom.mk_coe theorem coe_addMonoidHom_injective : Injective (fun f : α →+* β => (f : α →+ β)) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align ring_hom.coe_add_monoid_hom_injective RingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_monoidHom_injective : Injective (fun f : α →+* β => (f : α →* β)) := fun _ _ h => ext <| MonoidHom.congr_fun h #align ring_hom.coe_monoid_hom_injective RingHom.coe_monoidHom_injective /-- Ring homomorphisms map zero to zero. -/ protected theorem map_zero (f : α →+* β) : f 0 = 0 := map_zero f #align ring_hom.map_zero RingHom.map_zero /-- Ring homomorphisms map one to one. -/ protected theorem map_one (f : α →+* β) : f 1 = 1 := map_one f #align ring_hom.map_one RingHom.map_one /-- Ring homomorphisms preserve addition. -/ protected theorem map_add (f : α →+* β) : ∀ a b, f (a + b) = f a + f b := map_add f #align ring_hom.map_add RingHom.map_add /-- Ring homomorphisms preserve multiplication. -/ protected theorem map_mul (f : α →+* β) : ∀ a b, f (a * b) = f a * f b := map_mul f #align ring_hom.map_mul RingHom.map_mul @[simp] theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 0 1) = ite p 0 1 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_zero_one RingHom.map_ite_zero_one @[simp] theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 1 0) = ite p 1 0 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_one_zero RingHom.map_ite_one_zero /-- `f : α →+* β` has a trivial codomain iff `f 1 = 0`. -/ theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 := by rw [map_one, eq_comm] #align ring_hom.codomain_trivial_iff_map_one_eq_zero RingHom.codomain_trivial_iff_map_one_eq_zero /-- `f : α →+* β` has a trivial codomain iff it has a trivial range. -/ theorem codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 := f.codomain_trivial_iff_map_one_eq_zero.trans ⟨fun h x => by rw [← mul_one x, map_mul, h, mul_zero], fun h => h 1⟩ #align ring_hom.codomain_trivial_iff_range_trivial RingHom.codomain_trivial_iff_range_trivial /-- `f : α →+* β` doesn't map `1` to `0` if `β` is nontrivial -/ theorem map_one_ne_zero [Nontrivial β] : f 1 ≠ 0 := mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one #align ring_hom.map_one_ne_zero RingHom.map_one_ne_zero /-- If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. -/ theorem domain_nontrivial [Nontrivial β] : Nontrivial α := ⟨⟨1, 0, mt (fun h => show f 1 = 0 by rw [h, map_zero]) f.map_one_ne_zero⟩⟩ #align ring_hom.domain_nontrivial RingHom.domain_nontrivial theorem codomain_trivial (f : α →+* β) [h : Subsingleton α] : Subsingleton β := (subsingleton_or_nontrivial β).resolve_right fun _ => not_nontrivial_iff_subsingleton.mpr h f.domain_nontrivial #align ring_hom.codomain_trivial RingHom.codomain_trivial end /-- Ring homomorphisms preserve additive inverse. -/ protected theorem map_neg [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x : α) : f (-x) = -f x := map_neg f x #align ring_hom.map_neg RingHom.map_neg /-- Ring homomorphisms preserve subtraction. -/ protected theorem map_sub [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x y : α) : f (x - y) = f x - f y := map_sub f x y #align ring_hom.map_sub RingHom.map_sub /-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/ def mk' [NonAssocSemiring α] [NonAssocRing β] (f : α →* β) (map_add : ∀ a b, f (a + b) = f a + f b) : α →+* β := { AddMonoidHom.mk' f map_add, f with } #align ring_hom.mk' RingHom.mk' variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} /-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;> intros <;>

rfl

/-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;> intros <;>

Mathlib.Algebra.Ring.Hom.Defs.631_0.KyHvVYrIs9pW9ZQ

/-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α

Mathlib_Algebra_Ring_Hom_Defs

F : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 x✝² : NonAssocSemiring α x✝¹ : NonAssocSemiring β x✝ : NonAssocSemiring γ g : β →+* γ f : α →+* β src✝ : α →ₙ+* γ := NonUnitalRingHom.comp (toNonUnitalRingHom g) (toNonUnitalRingHom f) ⊢ (⇑g ∘ ⇑f) 1 = 1

/- Copyright (c) 2019 Amelia Livingston. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Amelia Livingston, Jireh Loreaux -/ import Mathlib.Algebra.Ring.Defs import Mathlib.Algebra.Ring.Basic import Mathlib.Data.Pi.Algebra #align_import algebra.hom.ring from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9" /-! # Homomorphisms of semirings and rings This file defines bundled homomorphisms of (non-unital) semirings and rings. As with monoid and groups, we use the same structure `RingHom a β`, a.k.a. `α →+* β`, for both types of homomorphisms. ## Main definitions * `NonUnitalRingHom`: Non-unital (semi)ring homomorphisms. Additive monoid homomorphism which preserve multiplication. * `RingHom`: (Semi)ring homomorphisms. Monoid homomorphisms which are also additive monoid homomorphism. ## Notations * `→ₙ+*`: Non-unital (semi)ring homs * `→+*`: (Semi)ring homs ## Implementation notes * There's a coercion from bundled homs to fun, and the canonical notation is to use the bundled hom as a function via this coercion. * There is no `SemiringHom` -- the idea is that `RingHom` is used. The constructor for a `RingHom` between semirings needs a proof of `map_zero`, `map_one` and `map_add` as well as `map_mul`; a separate constructor `RingHom.mk'` will construct ring homs between rings from monoid homs given only a proof that addition is preserved. ## Tags `RingHom`, `SemiringHom` -/ open Function variable {F α β γ : Type*} /-- Bundled non-unital semiring homomorphisms `α →ₙ+* β`; use this for bundled non-unital ring homomorphisms too. When possible, instead of parametrizing results over `(f : α →ₙ+* β)`, you should parametrize over `(F : Type*) [NonUnitalRingHomClass F α β] (f : F)`. When you extend this structure, make sure to extend `NonUnitalRingHomClass`. -/ structure NonUnitalRingHom (α β : Type*) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends α →ₙ* β, α →+ β #align non_unital_ring_hom NonUnitalRingHom /-- `α →ₙ+* β` denotes the type of non-unital ring homomorphisms from `α` to `β`. -/ infixr:25 " →ₙ+* " => NonUnitalRingHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as a semigroup homomorphism `α →ₙ* β`. The `simp`-normal form is `(f : α →ₙ* β)`. -/ add_decl_doc NonUnitalRingHom.toMulHom #align non_unital_ring_hom.to_mul_hom NonUnitalRingHom.toMulHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc NonUnitalRingHom.toAddMonoidHom #align non_unital_ring_hom.to_add_monoid_hom NonUnitalRingHom.toAddMonoidHom section NonUnitalRingHomClass /-- `NonUnitalRingHomClass F α β` states that `F` is a type of non-unital (semi)ring homomorphisms. You should extend this class when you extend `NonUnitalRingHom`. -/ class NonUnitalRingHomClass (F : Type*) (α β : outParam (Type*)) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends MulHomClass F α β, AddMonoidHomClass F α β #align non_unital_ring_hom_class NonUnitalRingHomClass variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] [NonUnitalRingHomClass F α β] /-- Turn an element of a type `F` satisfying `NonUnitalRingHomClass F α β` into an actual `NonUnitalRingHom`. This is declared as the default coercion from `F` to `α →ₙ+* β`. -/ @[coe] def NonUnitalRingHomClass.toNonUnitalRingHom (f : F) : α →ₙ+* β := { (f : α →ₙ* β), (f : α →+ β) with } /-- Any type satisfying `NonUnitalRingHomClass` can be cast into `NonUnitalRingHom` via `NonUnitalRingHomClass.toNonUnitalRingHom`. -/ instance : CoeTC F (α →ₙ+* β) := ⟨NonUnitalRingHomClass.toNonUnitalRingHom⟩ end NonUnitalRingHomClass namespace NonUnitalRingHom section coe variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] instance : NonUnitalRingHomClass (α →ₙ+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := NonUnitalRingHom.map_add' map_zero := NonUnitalRingHom.map_zero' map_mul f := f.map_mul' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -/ -- instance : CoeFun (α →ₙ+* β) fun _ => α → β := -- ⟨fun f => f.toFun⟩ -- Porting note: removed due to new `coe` in Lean4 #noalign non_unital_ring_hom.to_fun_eq_coe #noalign non_unital_ring_hom.coe_mk #noalign non_unital_ring_hom.coe_coe initialize_simps_projections NonUnitalRingHom (toFun → apply) @[simp] theorem coe_toMulHom (f : α →ₙ+* β) : ⇑f.toMulHom = f := rfl #align non_unital_ring_hom.coe_to_mul_hom NonUnitalRingHom.coe_toMulHom @[simp] theorem coe_mulHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →ₙ* β) = ⟨f, h₁⟩ := rfl #align non_unital_ring_hom.coe_mul_hom_mk NonUnitalRingHom.coe_mulHom_mk theorem coe_toAddMonoidHom (f : α →ₙ+* β) : ⇑f.toAddMonoidHom = f := rfl #align non_unital_ring_hom.coe_to_add_monoid_hom NonUnitalRingHom.coe_toAddMonoidHom @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →+ β) = ⟨⟨f, h₂⟩, h₃⟩ := rfl #align non_unital_ring_hom.coe_add_monoid_hom_mk NonUnitalRingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : α →ₙ+* β := { f.toMulHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align non_unital_ring_hom.copy NonUnitalRingHom.copy @[simp] theorem coe_copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align non_unital_ring_hom.coe_copy NonUnitalRingHom.coe_copy theorem copy_eq (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align non_unital_ring_hom.copy_eq NonUnitalRingHom.copy_eq end coe section variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] variable (f : α →ₙ+* β) {x y : α} @[ext] theorem ext ⦃f g : α →ₙ+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align non_unital_ring_hom.ext NonUnitalRingHom.ext theorem ext_iff {f g : α →ₙ+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align non_unital_ring_hom.ext_iff NonUnitalRingHom.ext_iff @[simp] theorem mk_coe (f : α →ₙ+* β) (h₁ h₂ h₃) : NonUnitalRingHom.mk (MulHom.mk f h₁) h₂ h₃ = f := ext fun _ => rfl #align non_unital_ring_hom.mk_coe NonUnitalRingHom.mk_coe theorem coe_addMonoidHom_injective : Injective fun f : α →ₙ+* β => (f : α →+ β) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align non_unital_ring_hom.coe_add_monoid_hom_injective NonUnitalRingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_mulHom_injective : Injective fun f : α →ₙ+* β => (f : α →ₙ* β) := fun _ _ h => ext <| MulHom.congr_fun h #align non_unital_ring_hom.coe_mul_hom_injective NonUnitalRingHom.coe_mulHom_injective end variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/ protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by refine' { toFun := id.. } <;> intros <;> rfl #align non_unital_ring_hom.id NonUnitalRingHom.id instance : Zero (α →ₙ+* β) := ⟨{ toFun := 0, map_mul' := fun _ _ => (mul_zero (0 : β)).symm, map_zero' := rfl, map_add' := fun _ _ => (add_zero (0 : β)).symm }⟩ instance : Inhabited (α →ₙ+* β) := ⟨0⟩ @[simp] theorem coe_zero : ⇑(0 : α →ₙ+* β) = 0 := rfl #align non_unital_ring_hom.coe_zero NonUnitalRingHom.coe_zero @[simp] theorem zero_apply (x : α) : (0 : α →ₙ+* β) x = 0 := rfl #align non_unital_ring_hom.zero_apply NonUnitalRingHom.zero_apply @[simp] theorem id_apply (x : α) : NonUnitalRingHom.id α x = x := rfl #align non_unital_ring_hom.id_apply NonUnitalRingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (NonUnitalRingHom.id α : α →+ α) = AddMonoidHom.id α := rfl #align non_unital_ring_hom.coe_add_monoid_hom_id NonUnitalRingHom.coe_addMonoidHom_id @[simp] theorem coe_mulHom_id : (NonUnitalRingHom.id α : α →ₙ* α) = MulHom.id α := rfl #align non_unital_ring_hom.coe_mul_hom_id NonUnitalRingHom.coe_mulHom_id variable [NonUnitalNonAssocSemiring γ] /-- Composition of non-unital ring homomorphisms is a non-unital ring homomorphism. -/ def comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : α →ₙ+* γ := { g.toMulHom.comp f.toMulHom, g.toAddMonoidHom.comp f.toAddMonoidHom with } #align non_unital_ring_hom.comp NonUnitalRingHom.comp /-- Composition of non-unital ring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonUnitalNonAssocSemiring δ} (f : α →ₙ+* β) (g : β →ₙ+* γ) (h : γ →ₙ+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align non_unital_ring_hom.comp_assoc NonUnitalRingHom.comp_assoc @[simp] theorem coe_comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : ⇑(g.comp f) = g ∘ f := rfl #align non_unital_ring_hom.coe_comp NonUnitalRingHom.coe_comp @[simp] theorem comp_apply (g : β →ₙ+* γ) (f : α →ₙ+* β) (x : α) : g.comp f x = g (f x) := rfl #align non_unital_ring_hom.comp_apply NonUnitalRingHom.comp_apply variable (g : β →ₙ+* γ) (f : α →ₙ+* β) @[simp] theorem coe_comp_addMonoidHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : AddMonoidHom.mk ⟨g ∘ f, (g.comp f).map_zero'⟩ (g.comp f).map_add' = (g : β →+ γ).comp f := rfl #align non_unital_ring_hom.coe_comp_add_monoid_hom NonUnitalRingHom.coe_comp_addMonoidHom @[simp] theorem coe_comp_mulHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : MulHom.mk (g ∘ f) (g.comp f).map_mul' = (g : β →ₙ* γ).comp f := rfl #align non_unital_ring_hom.coe_comp_mul_hom NonUnitalRingHom.coe_comp_mulHom @[simp] theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 := by ext simp #align non_unital_ring_hom.comp_zero NonUnitalRingHom.comp_zero @[simp] theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 := by ext rfl #align non_unital_ring_hom.zero_comp NonUnitalRingHom.zero_comp @[simp] theorem comp_id (f : α →ₙ+* β) : f.comp (NonUnitalRingHom.id α) = f := ext fun _ => rfl #align non_unital_ring_hom.comp_id NonUnitalRingHom.comp_id @[simp] theorem id_comp (f : α →ₙ+* β) : (NonUnitalRingHom.id β).comp f = f := ext fun _ => rfl #align non_unital_ring_hom.id_comp NonUnitalRingHom.id_comp instance : MonoidWithZero (α →ₙ+* α) where one := NonUnitalRingHom.id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ zero := 0 mul_zero := comp_zero zero_mul := zero_comp theorem one_def : (1 : α →ₙ+* α) = NonUnitalRingHom.id α := rfl #align non_unital_ring_hom.one_def NonUnitalRingHom.one_def @[simp] theorem coe_one : ⇑(1 : α →ₙ+* α) = id := rfl #align non_unital_ring_hom.coe_one NonUnitalRingHom.coe_one theorem mul_def (f g : α →ₙ+* α) : f * g = f.comp g := rfl #align non_unital_ring_hom.mul_def NonUnitalRingHom.mul_def @[simp] theorem coe_mul (f g : α →ₙ+* α) : ⇑(f * g) = f ∘ g := rfl #align non_unital_ring_hom.coe_mul NonUnitalRingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →ₙ+* γ} {f : α →ₙ+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_right NonUnitalRingHom.cancel_right @[simp] theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_left NonUnitalRingHom.cancel_left end NonUnitalRingHom /-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too. This extends from both `MonoidHom` and `MonoidWithZeroHom` in order to put the fields in a sensible order, even though `MonoidWithZeroHom` already extends `MonoidHom`. -/ structure RingHom (α : Type*) (β : Type*) [NonAssocSemiring α] [NonAssocSemiring β] extends α →* β, α →+ β, α →ₙ+* β, α →*₀ β #align ring_hom RingHom /-- `α →+* β` denotes the type of ring homomorphisms from `α` to `β`. -/ infixr:25 " →+* " => RingHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid with zero homomorphism `α →*₀ β`. The `simp`-normal form is `(f : α →*₀ β)`. -/ add_decl_doc RingHom.toMonoidWithZeroHom #align ring_hom.to_monoid_with_zero_hom RingHom.toMonoidWithZeroHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid homomorphism `α →* β`. The `simp`-normal form is `(f : α →* β)`. -/ add_decl_doc RingHom.toMonoidHom #align ring_hom.to_monoid_hom RingHom.toMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc RingHom.toAddMonoidHom #align ring_hom.to_add_monoid_hom RingHom.toAddMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a non-unital ring homomorphism `α →ₙ+* β`. The `simp`-normal form is `(f : α →ₙ+* β)`. -/ add_decl_doc RingHom.toNonUnitalRingHom #align ring_hom.to_non_unital_ring_hom RingHom.toNonUnitalRingHom section RingHomClass /-- `RingHomClass F α β` states that `F` is a type of (semi)ring homomorphisms. You should extend this class when you extend `RingHom`. This extends from both `MonoidHomClass` and `MonoidWithZeroHomClass` in order to put the fields in a sensible order, even though `MonoidWithZeroHomClass` already extends `MonoidHomClass`. -/ class RingHomClass (F : Type*) (α β : outParam (Type*)) [NonAssocSemiring α] [NonAssocSemiring β] extends MonoidHomClass F α β, AddMonoidHomClass F α β, MonoidWithZeroHomClass F α β #align ring_hom_class RingHomClass set_option linter.deprecated false in /-- Ring homomorphisms preserve `bit1`. -/ @[simp] lemma map_bit1 [NonAssocSemiring α] [NonAssocSemiring β] [RingHomClass F α β] (f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) := by simp [bit1] #align map_bit1 map_bit1 -- Porting note: marked `{}` rather than `[]` to prevent dangerous instances variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} [RingHomClass F α β] /-- Turn an element of a type `F` satisfying `RingHomClass F α β` into an actual `RingHom`. This is declared as the default coercion from `F` to `α →+* β`. -/ @[coe] def RingHomClass.toRingHom (f : F) : α →+* β := { (f : α →* β), (f : α →+ β) with } /-- Any type satisfying `RingHomClass` can be cast into `RingHom` via `RingHomClass.toRingHom`. -/ instance : CoeTC F (α →+* β) := ⟨RingHomClass.toRingHom⟩ instance (priority := 100) RingHomClass.toNonUnitalRingHomClass : NonUnitalRingHomClass F α β := { ‹RingHomClass F α β› with } #align ring_hom_class.to_non_unital_ring_hom_class RingHomClass.toNonUnitalRingHomClass end RingHomClass namespace RingHom section coe /-! Throughout this section, some `Semiring` arguments are specified with `{}` instead of `[]`. See note [implicit instance arguments]. -/ variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} instance instRingHomClass : RingHomClass (α →+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := RingHom.map_add' map_zero := RingHom.map_zero' map_mul f := f.map_mul' map_one f := f.map_one' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -- -/ -- instance : CoeFun (α →+* β) fun _ => α → β := -- ⟨RingHom.toFun⟩ initialize_simps_projections RingHom (toFun → apply) -- Porting note: is this lemma still needed in Lean4? -- Porting note: because `f.toFun` really means `f.toMonoidHom.toOneHom.toFun` and -- `toMonoidHom_eq_coe` wants to simplify `f.toMonoidHom` to `(↑f : M →* N)`, this can't -- be a simp lemma anymore -- @[simp] theorem toFun_eq_coe (f : α →+* β) : f.toFun = f := rfl #align ring_hom.to_fun_eq_coe RingHom.toFun_eq_coe @[simp] theorem coe_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α → β) = f := rfl #align ring_hom.coe_mk RingHom.coe_mk @[simp] theorem coe_coe {F : Type*} [RingHomClass F α β] (f : F) : ((f : α →+* β) : α → β) = f := rfl #align ring_hom.coe_coe RingHom.coe_coe attribute [coe] RingHom.toMonoidHom instance coeToMonoidHom : Coe (α →+* β) (α →* β) := ⟨RingHom.toMonoidHom⟩ #align ring_hom.has_coe_monoid_hom RingHom.coeToMonoidHom -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_monoid_hom @[simp] theorem toMonoidHom_eq_coe (f : α →+* β) : f.toMonoidHom = f := rfl #align ring_hom.to_monoid_hom_eq_coe RingHom.toMonoidHom_eq_coe -- Porting note: this can't be a simp lemma anymore -- @[simp] theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f := by rfl #align ring_hom.to_monoid_with_zero_hom_eq_coe RingHom.toMonoidWithZeroHom_eq_coe @[simp] theorem coe_monoidHom_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α →* β) = f := rfl #align ring_hom.coe_monoid_hom_mk RingHom.coe_monoidHom_mk -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_add_monoid_hom @[simp] theorem toAddMonoidHom_eq_coe (f : α →+* β) : f.toAddMonoidHom = f := rfl #align ring_hom.to_add_monoid_hom_eq_coe RingHom.toAddMonoidHom_eq_coe @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃ h₄) : ((⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨⟨f, h₃⟩, h₄⟩ := rfl #align ring_hom.coe_add_monoid_hom_mk RingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ def copy (f : α →+* β) (f' : α → β) (h : f' = f) : α →+* β := { f.toMonoidWithZeroHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align ring_hom.copy RingHom.copy @[simp] theorem coe_copy (f : α →+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align ring_hom.coe_copy RingHom.coe_copy theorem copy_eq (f : α →+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align ring_hom.copy_eq RingHom.copy_eq end coe section variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} (f : α →+* β) {x y : α} theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x := FunLike.congr_fun h x #align ring_hom.congr_fun RingHom.congr_fun theorem congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y := FunLike.congr_arg f h #align ring_hom.congr_arg RingHom.congr_arg theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g := FunLike.coe_injective h #align ring_hom.coe_inj RingHom.coe_inj @[ext] theorem ext ⦃f g : α →+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align ring_hom.ext RingHom.ext theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align ring_hom.ext_iff RingHom.ext_iff @[simp] theorem mk_coe (f : α →+* β) (h₁ h₂ h₃ h₄) : RingHom.mk ⟨⟨f, h₁⟩, h₂⟩ h₃ h₄ = f := ext fun _ => rfl #align ring_hom.mk_coe RingHom.mk_coe theorem coe_addMonoidHom_injective : Injective (fun f : α →+* β => (f : α →+ β)) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align ring_hom.coe_add_monoid_hom_injective RingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_monoidHom_injective : Injective (fun f : α →+* β => (f : α →* β)) := fun _ _ h => ext <| MonoidHom.congr_fun h #align ring_hom.coe_monoid_hom_injective RingHom.coe_monoidHom_injective /-- Ring homomorphisms map zero to zero. -/ protected theorem map_zero (f : α →+* β) : f 0 = 0 := map_zero f #align ring_hom.map_zero RingHom.map_zero /-- Ring homomorphisms map one to one. -/ protected theorem map_one (f : α →+* β) : f 1 = 1 := map_one f #align ring_hom.map_one RingHom.map_one /-- Ring homomorphisms preserve addition. -/ protected theorem map_add (f : α →+* β) : ∀ a b, f (a + b) = f a + f b := map_add f #align ring_hom.map_add RingHom.map_add /-- Ring homomorphisms preserve multiplication. -/ protected theorem map_mul (f : α →+* β) : ∀ a b, f (a * b) = f a * f b := map_mul f #align ring_hom.map_mul RingHom.map_mul @[simp] theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 0 1) = ite p 0 1 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_zero_one RingHom.map_ite_zero_one @[simp] theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 1 0) = ite p 1 0 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_one_zero RingHom.map_ite_one_zero /-- `f : α →+* β` has a trivial codomain iff `f 1 = 0`. -/ theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 := by rw [map_one, eq_comm] #align ring_hom.codomain_trivial_iff_map_one_eq_zero RingHom.codomain_trivial_iff_map_one_eq_zero /-- `f : α →+* β` has a trivial codomain iff it has a trivial range. -/ theorem codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 := f.codomain_trivial_iff_map_one_eq_zero.trans ⟨fun h x => by rw [← mul_one x, map_mul, h, mul_zero], fun h => h 1⟩ #align ring_hom.codomain_trivial_iff_range_trivial RingHom.codomain_trivial_iff_range_trivial /-- `f : α →+* β` doesn't map `1` to `0` if `β` is nontrivial -/ theorem map_one_ne_zero [Nontrivial β] : f 1 ≠ 0 := mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one #align ring_hom.map_one_ne_zero RingHom.map_one_ne_zero /-- If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. -/ theorem domain_nontrivial [Nontrivial β] : Nontrivial α := ⟨⟨1, 0, mt (fun h => show f 1 = 0 by rw [h, map_zero]) f.map_one_ne_zero⟩⟩ #align ring_hom.domain_nontrivial RingHom.domain_nontrivial theorem codomain_trivial (f : α →+* β) [h : Subsingleton α] : Subsingleton β := (subsingleton_or_nontrivial β).resolve_right fun _ => not_nontrivial_iff_subsingleton.mpr h f.domain_nontrivial #align ring_hom.codomain_trivial RingHom.codomain_trivial end /-- Ring homomorphisms preserve additive inverse. -/ protected theorem map_neg [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x : α) : f (-x) = -f x := map_neg f x #align ring_hom.map_neg RingHom.map_neg /-- Ring homomorphisms preserve subtraction. -/ protected theorem map_sub [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x y : α) : f (x - y) = f x - f y := map_sub f x y #align ring_hom.map_sub RingHom.map_sub /-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/ def mk' [NonAssocSemiring α] [NonAssocRing β] (f : α →* β) (map_add : ∀ a b, f (a + b) = f a + f b) : α →+* β := { AddMonoidHom.mk' f map_add, f with } #align ring_hom.mk' RingHom.mk' variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} /-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;> intros <;> rfl #align ring_hom.id RingHom.id instance : Inhabited (α →+* α) := ⟨id α⟩ @[simp] theorem id_apply (x : α) : RingHom.id α x = x := rfl #align ring_hom.id_apply RingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (id α : α →+ α) = AddMonoidHom.id α := rfl #align ring_hom.coe_add_monoid_hom_id RingHom.coe_addMonoidHom_id @[simp] theorem coe_monoidHom_id : (id α : α →* α) = MonoidHom.id α := rfl #align ring_hom.coe_monoid_hom_id RingHom.coe_monoidHom_id variable {_ : NonAssocSemiring γ} /-- Composition of ring homomorphisms is a ring homomorphism. -/ def comp (g : β →+* γ) (f : α →+* β) : α →+* γ := { g.toNonUnitalRingHom.comp f.toNonUnitalRingHom with toFun := g ∘ f, map_one' := by

simp

/-- Composition of ring homomorphisms is a ring homomorphism. -/ def comp (g : β →+* γ) (f : α →+* β) : α →+* γ := { g.toNonUnitalRingHom.comp f.toNonUnitalRingHom with toFun := g ∘ f, map_one' := by

Mathlib.Algebra.Ring.Hom.Defs.656_0.KyHvVYrIs9pW9ZQ

/-- Composition of ring homomorphisms is a ring homomorphism. -/ def comp (g : β →+* γ) (f : α →+* β) : α →+* γ

Mathlib_Algebra_Ring_Hom_Defs

F : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 x✝² : NonAssocSemiring α x✝¹ : NonAssocSemiring β x✝ : NonAssocSemiring γ g : β →+* γ f₁ f₂ : α →+* β hg : Injective ⇑g h : comp g f₁ = comp g f₂ x : α ⊢ g (f₁ x) = g (f₂ x)

/- Copyright (c) 2019 Amelia Livingston. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Amelia Livingston, Jireh Loreaux -/ import Mathlib.Algebra.Ring.Defs import Mathlib.Algebra.Ring.Basic import Mathlib.Data.Pi.Algebra #align_import algebra.hom.ring from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9" /-! # Homomorphisms of semirings and rings This file defines bundled homomorphisms of (non-unital) semirings and rings. As with monoid and groups, we use the same structure `RingHom a β`, a.k.a. `α →+* β`, for both types of homomorphisms. ## Main definitions * `NonUnitalRingHom`: Non-unital (semi)ring homomorphisms. Additive monoid homomorphism which preserve multiplication. * `RingHom`: (Semi)ring homomorphisms. Monoid homomorphisms which are also additive monoid homomorphism. ## Notations * `→ₙ+*`: Non-unital (semi)ring homs * `→+*`: (Semi)ring homs ## Implementation notes * There's a coercion from bundled homs to fun, and the canonical notation is to use the bundled hom as a function via this coercion. * There is no `SemiringHom` -- the idea is that `RingHom` is used. The constructor for a `RingHom` between semirings needs a proof of `map_zero`, `map_one` and `map_add` as well as `map_mul`; a separate constructor `RingHom.mk'` will construct ring homs between rings from monoid homs given only a proof that addition is preserved. ## Tags `RingHom`, `SemiringHom` -/ open Function variable {F α β γ : Type*} /-- Bundled non-unital semiring homomorphisms `α →ₙ+* β`; use this for bundled non-unital ring homomorphisms too. When possible, instead of parametrizing results over `(f : α →ₙ+* β)`, you should parametrize over `(F : Type*) [NonUnitalRingHomClass F α β] (f : F)`. When you extend this structure, make sure to extend `NonUnitalRingHomClass`. -/ structure NonUnitalRingHom (α β : Type*) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends α →ₙ* β, α →+ β #align non_unital_ring_hom NonUnitalRingHom /-- `α →ₙ+* β` denotes the type of non-unital ring homomorphisms from `α` to `β`. -/ infixr:25 " →ₙ+* " => NonUnitalRingHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as a semigroup homomorphism `α →ₙ* β`. The `simp`-normal form is `(f : α →ₙ* β)`. -/ add_decl_doc NonUnitalRingHom.toMulHom #align non_unital_ring_hom.to_mul_hom NonUnitalRingHom.toMulHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc NonUnitalRingHom.toAddMonoidHom #align non_unital_ring_hom.to_add_monoid_hom NonUnitalRingHom.toAddMonoidHom section NonUnitalRingHomClass /-- `NonUnitalRingHomClass F α β` states that `F` is a type of non-unital (semi)ring homomorphisms. You should extend this class when you extend `NonUnitalRingHom`. -/ class NonUnitalRingHomClass (F : Type*) (α β : outParam (Type*)) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends MulHomClass F α β, AddMonoidHomClass F α β #align non_unital_ring_hom_class NonUnitalRingHomClass variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] [NonUnitalRingHomClass F α β] /-- Turn an element of a type `F` satisfying `NonUnitalRingHomClass F α β` into an actual `NonUnitalRingHom`. This is declared as the default coercion from `F` to `α →ₙ+* β`. -/ @[coe] def NonUnitalRingHomClass.toNonUnitalRingHom (f : F) : α →ₙ+* β := { (f : α →ₙ* β), (f : α →+ β) with } /-- Any type satisfying `NonUnitalRingHomClass` can be cast into `NonUnitalRingHom` via `NonUnitalRingHomClass.toNonUnitalRingHom`. -/ instance : CoeTC F (α →ₙ+* β) := ⟨NonUnitalRingHomClass.toNonUnitalRingHom⟩ end NonUnitalRingHomClass namespace NonUnitalRingHom section coe variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] instance : NonUnitalRingHomClass (α →ₙ+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := NonUnitalRingHom.map_add' map_zero := NonUnitalRingHom.map_zero' map_mul f := f.map_mul' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -/ -- instance : CoeFun (α →ₙ+* β) fun _ => α → β := -- ⟨fun f => f.toFun⟩ -- Porting note: removed due to new `coe` in Lean4 #noalign non_unital_ring_hom.to_fun_eq_coe #noalign non_unital_ring_hom.coe_mk #noalign non_unital_ring_hom.coe_coe initialize_simps_projections NonUnitalRingHom (toFun → apply) @[simp] theorem coe_toMulHom (f : α →ₙ+* β) : ⇑f.toMulHom = f := rfl #align non_unital_ring_hom.coe_to_mul_hom NonUnitalRingHom.coe_toMulHom @[simp] theorem coe_mulHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →ₙ* β) = ⟨f, h₁⟩ := rfl #align non_unital_ring_hom.coe_mul_hom_mk NonUnitalRingHom.coe_mulHom_mk theorem coe_toAddMonoidHom (f : α →ₙ+* β) : ⇑f.toAddMonoidHom = f := rfl #align non_unital_ring_hom.coe_to_add_monoid_hom NonUnitalRingHom.coe_toAddMonoidHom @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →+ β) = ⟨⟨f, h₂⟩, h₃⟩ := rfl #align non_unital_ring_hom.coe_add_monoid_hom_mk NonUnitalRingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : α →ₙ+* β := { f.toMulHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align non_unital_ring_hom.copy NonUnitalRingHom.copy @[simp] theorem coe_copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align non_unital_ring_hom.coe_copy NonUnitalRingHom.coe_copy theorem copy_eq (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align non_unital_ring_hom.copy_eq NonUnitalRingHom.copy_eq end coe section variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] variable (f : α →ₙ+* β) {x y : α} @[ext] theorem ext ⦃f g : α →ₙ+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align non_unital_ring_hom.ext NonUnitalRingHom.ext theorem ext_iff {f g : α →ₙ+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align non_unital_ring_hom.ext_iff NonUnitalRingHom.ext_iff @[simp] theorem mk_coe (f : α →ₙ+* β) (h₁ h₂ h₃) : NonUnitalRingHom.mk (MulHom.mk f h₁) h₂ h₃ = f := ext fun _ => rfl #align non_unital_ring_hom.mk_coe NonUnitalRingHom.mk_coe theorem coe_addMonoidHom_injective : Injective fun f : α →ₙ+* β => (f : α →+ β) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align non_unital_ring_hom.coe_add_monoid_hom_injective NonUnitalRingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_mulHom_injective : Injective fun f : α →ₙ+* β => (f : α →ₙ* β) := fun _ _ h => ext <| MulHom.congr_fun h #align non_unital_ring_hom.coe_mul_hom_injective NonUnitalRingHom.coe_mulHom_injective end variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/ protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by refine' { toFun := id.. } <;> intros <;> rfl #align non_unital_ring_hom.id NonUnitalRingHom.id instance : Zero (α →ₙ+* β) := ⟨{ toFun := 0, map_mul' := fun _ _ => (mul_zero (0 : β)).symm, map_zero' := rfl, map_add' := fun _ _ => (add_zero (0 : β)).symm }⟩ instance : Inhabited (α →ₙ+* β) := ⟨0⟩ @[simp] theorem coe_zero : ⇑(0 : α →ₙ+* β) = 0 := rfl #align non_unital_ring_hom.coe_zero NonUnitalRingHom.coe_zero @[simp] theorem zero_apply (x : α) : (0 : α →ₙ+* β) x = 0 := rfl #align non_unital_ring_hom.zero_apply NonUnitalRingHom.zero_apply @[simp] theorem id_apply (x : α) : NonUnitalRingHom.id α x = x := rfl #align non_unital_ring_hom.id_apply NonUnitalRingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (NonUnitalRingHom.id α : α →+ α) = AddMonoidHom.id α := rfl #align non_unital_ring_hom.coe_add_monoid_hom_id NonUnitalRingHom.coe_addMonoidHom_id @[simp] theorem coe_mulHom_id : (NonUnitalRingHom.id α : α →ₙ* α) = MulHom.id α := rfl #align non_unital_ring_hom.coe_mul_hom_id NonUnitalRingHom.coe_mulHom_id variable [NonUnitalNonAssocSemiring γ] /-- Composition of non-unital ring homomorphisms is a non-unital ring homomorphism. -/ def comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : α →ₙ+* γ := { g.toMulHom.comp f.toMulHom, g.toAddMonoidHom.comp f.toAddMonoidHom with } #align non_unital_ring_hom.comp NonUnitalRingHom.comp /-- Composition of non-unital ring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonUnitalNonAssocSemiring δ} (f : α →ₙ+* β) (g : β →ₙ+* γ) (h : γ →ₙ+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align non_unital_ring_hom.comp_assoc NonUnitalRingHom.comp_assoc @[simp] theorem coe_comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : ⇑(g.comp f) = g ∘ f := rfl #align non_unital_ring_hom.coe_comp NonUnitalRingHom.coe_comp @[simp] theorem comp_apply (g : β →ₙ+* γ) (f : α →ₙ+* β) (x : α) : g.comp f x = g (f x) := rfl #align non_unital_ring_hom.comp_apply NonUnitalRingHom.comp_apply variable (g : β →ₙ+* γ) (f : α →ₙ+* β) @[simp] theorem coe_comp_addMonoidHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : AddMonoidHom.mk ⟨g ∘ f, (g.comp f).map_zero'⟩ (g.comp f).map_add' = (g : β →+ γ).comp f := rfl #align non_unital_ring_hom.coe_comp_add_monoid_hom NonUnitalRingHom.coe_comp_addMonoidHom @[simp] theorem coe_comp_mulHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : MulHom.mk (g ∘ f) (g.comp f).map_mul' = (g : β →ₙ* γ).comp f := rfl #align non_unital_ring_hom.coe_comp_mul_hom NonUnitalRingHom.coe_comp_mulHom @[simp] theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 := by ext simp #align non_unital_ring_hom.comp_zero NonUnitalRingHom.comp_zero @[simp] theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 := by ext rfl #align non_unital_ring_hom.zero_comp NonUnitalRingHom.zero_comp @[simp] theorem comp_id (f : α →ₙ+* β) : f.comp (NonUnitalRingHom.id α) = f := ext fun _ => rfl #align non_unital_ring_hom.comp_id NonUnitalRingHom.comp_id @[simp] theorem id_comp (f : α →ₙ+* β) : (NonUnitalRingHom.id β).comp f = f := ext fun _ => rfl #align non_unital_ring_hom.id_comp NonUnitalRingHom.id_comp instance : MonoidWithZero (α →ₙ+* α) where one := NonUnitalRingHom.id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ zero := 0 mul_zero := comp_zero zero_mul := zero_comp theorem one_def : (1 : α →ₙ+* α) = NonUnitalRingHom.id α := rfl #align non_unital_ring_hom.one_def NonUnitalRingHom.one_def @[simp] theorem coe_one : ⇑(1 : α →ₙ+* α) = id := rfl #align non_unital_ring_hom.coe_one NonUnitalRingHom.coe_one theorem mul_def (f g : α →ₙ+* α) : f * g = f.comp g := rfl #align non_unital_ring_hom.mul_def NonUnitalRingHom.mul_def @[simp] theorem coe_mul (f g : α →ₙ+* α) : ⇑(f * g) = f ∘ g := rfl #align non_unital_ring_hom.coe_mul NonUnitalRingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →ₙ+* γ} {f : α →ₙ+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_right NonUnitalRingHom.cancel_right @[simp] theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_left NonUnitalRingHom.cancel_left end NonUnitalRingHom /-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too. This extends from both `MonoidHom` and `MonoidWithZeroHom` in order to put the fields in a sensible order, even though `MonoidWithZeroHom` already extends `MonoidHom`. -/ structure RingHom (α : Type*) (β : Type*) [NonAssocSemiring α] [NonAssocSemiring β] extends α →* β, α →+ β, α →ₙ+* β, α →*₀ β #align ring_hom RingHom /-- `α →+* β` denotes the type of ring homomorphisms from `α` to `β`. -/ infixr:25 " →+* " => RingHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid with zero homomorphism `α →*₀ β`. The `simp`-normal form is `(f : α →*₀ β)`. -/ add_decl_doc RingHom.toMonoidWithZeroHom #align ring_hom.to_monoid_with_zero_hom RingHom.toMonoidWithZeroHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid homomorphism `α →* β`. The `simp`-normal form is `(f : α →* β)`. -/ add_decl_doc RingHom.toMonoidHom #align ring_hom.to_monoid_hom RingHom.toMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc RingHom.toAddMonoidHom #align ring_hom.to_add_monoid_hom RingHom.toAddMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a non-unital ring homomorphism `α →ₙ+* β`. The `simp`-normal form is `(f : α →ₙ+* β)`. -/ add_decl_doc RingHom.toNonUnitalRingHom #align ring_hom.to_non_unital_ring_hom RingHom.toNonUnitalRingHom section RingHomClass /-- `RingHomClass F α β` states that `F` is a type of (semi)ring homomorphisms. You should extend this class when you extend `RingHom`. This extends from both `MonoidHomClass` and `MonoidWithZeroHomClass` in order to put the fields in a sensible order, even though `MonoidWithZeroHomClass` already extends `MonoidHomClass`. -/ class RingHomClass (F : Type*) (α β : outParam (Type*)) [NonAssocSemiring α] [NonAssocSemiring β] extends MonoidHomClass F α β, AddMonoidHomClass F α β, MonoidWithZeroHomClass F α β #align ring_hom_class RingHomClass set_option linter.deprecated false in /-- Ring homomorphisms preserve `bit1`. -/ @[simp] lemma map_bit1 [NonAssocSemiring α] [NonAssocSemiring β] [RingHomClass F α β] (f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) := by simp [bit1] #align map_bit1 map_bit1 -- Porting note: marked `{}` rather than `[]` to prevent dangerous instances variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} [RingHomClass F α β] /-- Turn an element of a type `F` satisfying `RingHomClass F α β` into an actual `RingHom`. This is declared as the default coercion from `F` to `α →+* β`. -/ @[coe] def RingHomClass.toRingHom (f : F) : α →+* β := { (f : α →* β), (f : α →+ β) with } /-- Any type satisfying `RingHomClass` can be cast into `RingHom` via `RingHomClass.toRingHom`. -/ instance : CoeTC F (α →+* β) := ⟨RingHomClass.toRingHom⟩ instance (priority := 100) RingHomClass.toNonUnitalRingHomClass : NonUnitalRingHomClass F α β := { ‹RingHomClass F α β› with } #align ring_hom_class.to_non_unital_ring_hom_class RingHomClass.toNonUnitalRingHomClass end RingHomClass namespace RingHom section coe /-! Throughout this section, some `Semiring` arguments are specified with `{}` instead of `[]`. See note [implicit instance arguments]. -/ variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} instance instRingHomClass : RingHomClass (α →+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := RingHom.map_add' map_zero := RingHom.map_zero' map_mul f := f.map_mul' map_one f := f.map_one' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -- -/ -- instance : CoeFun (α →+* β) fun _ => α → β := -- ⟨RingHom.toFun⟩ initialize_simps_projections RingHom (toFun → apply) -- Porting note: is this lemma still needed in Lean4? -- Porting note: because `f.toFun` really means `f.toMonoidHom.toOneHom.toFun` and -- `toMonoidHom_eq_coe` wants to simplify `f.toMonoidHom` to `(↑f : M →* N)`, this can't -- be a simp lemma anymore -- @[simp] theorem toFun_eq_coe (f : α →+* β) : f.toFun = f := rfl #align ring_hom.to_fun_eq_coe RingHom.toFun_eq_coe @[simp] theorem coe_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α → β) = f := rfl #align ring_hom.coe_mk RingHom.coe_mk @[simp] theorem coe_coe {F : Type*} [RingHomClass F α β] (f : F) : ((f : α →+* β) : α → β) = f := rfl #align ring_hom.coe_coe RingHom.coe_coe attribute [coe] RingHom.toMonoidHom instance coeToMonoidHom : Coe (α →+* β) (α →* β) := ⟨RingHom.toMonoidHom⟩ #align ring_hom.has_coe_monoid_hom RingHom.coeToMonoidHom -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_monoid_hom @[simp] theorem toMonoidHom_eq_coe (f : α →+* β) : f.toMonoidHom = f := rfl #align ring_hom.to_monoid_hom_eq_coe RingHom.toMonoidHom_eq_coe -- Porting note: this can't be a simp lemma anymore -- @[simp] theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f := by rfl #align ring_hom.to_monoid_with_zero_hom_eq_coe RingHom.toMonoidWithZeroHom_eq_coe @[simp] theorem coe_monoidHom_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α →* β) = f := rfl #align ring_hom.coe_monoid_hom_mk RingHom.coe_monoidHom_mk -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_add_monoid_hom @[simp] theorem toAddMonoidHom_eq_coe (f : α →+* β) : f.toAddMonoidHom = f := rfl #align ring_hom.to_add_monoid_hom_eq_coe RingHom.toAddMonoidHom_eq_coe @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃ h₄) : ((⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨⟨f, h₃⟩, h₄⟩ := rfl #align ring_hom.coe_add_monoid_hom_mk RingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ def copy (f : α →+* β) (f' : α → β) (h : f' = f) : α →+* β := { f.toMonoidWithZeroHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align ring_hom.copy RingHom.copy @[simp] theorem coe_copy (f : α →+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align ring_hom.coe_copy RingHom.coe_copy theorem copy_eq (f : α →+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align ring_hom.copy_eq RingHom.copy_eq end coe section variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} (f : α →+* β) {x y : α} theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x := FunLike.congr_fun h x #align ring_hom.congr_fun RingHom.congr_fun theorem congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y := FunLike.congr_arg f h #align ring_hom.congr_arg RingHom.congr_arg theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g := FunLike.coe_injective h #align ring_hom.coe_inj RingHom.coe_inj @[ext] theorem ext ⦃f g : α →+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align ring_hom.ext RingHom.ext theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align ring_hom.ext_iff RingHom.ext_iff @[simp] theorem mk_coe (f : α →+* β) (h₁ h₂ h₃ h₄) : RingHom.mk ⟨⟨f, h₁⟩, h₂⟩ h₃ h₄ = f := ext fun _ => rfl #align ring_hom.mk_coe RingHom.mk_coe theorem coe_addMonoidHom_injective : Injective (fun f : α →+* β => (f : α →+ β)) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align ring_hom.coe_add_monoid_hom_injective RingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_monoidHom_injective : Injective (fun f : α →+* β => (f : α →* β)) := fun _ _ h => ext <| MonoidHom.congr_fun h #align ring_hom.coe_monoid_hom_injective RingHom.coe_monoidHom_injective /-- Ring homomorphisms map zero to zero. -/ protected theorem map_zero (f : α →+* β) : f 0 = 0 := map_zero f #align ring_hom.map_zero RingHom.map_zero /-- Ring homomorphisms map one to one. -/ protected theorem map_one (f : α →+* β) : f 1 = 1 := map_one f #align ring_hom.map_one RingHom.map_one /-- Ring homomorphisms preserve addition. -/ protected theorem map_add (f : α →+* β) : ∀ a b, f (a + b) = f a + f b := map_add f #align ring_hom.map_add RingHom.map_add /-- Ring homomorphisms preserve multiplication. -/ protected theorem map_mul (f : α →+* β) : ∀ a b, f (a * b) = f a * f b := map_mul f #align ring_hom.map_mul RingHom.map_mul @[simp] theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 0 1) = ite p 0 1 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_zero_one RingHom.map_ite_zero_one @[simp] theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 1 0) = ite p 1 0 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_one_zero RingHom.map_ite_one_zero /-- `f : α →+* β` has a trivial codomain iff `f 1 = 0`. -/ theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 := by rw [map_one, eq_comm] #align ring_hom.codomain_trivial_iff_map_one_eq_zero RingHom.codomain_trivial_iff_map_one_eq_zero /-- `f : α →+* β` has a trivial codomain iff it has a trivial range. -/ theorem codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 := f.codomain_trivial_iff_map_one_eq_zero.trans ⟨fun h x => by rw [← mul_one x, map_mul, h, mul_zero], fun h => h 1⟩ #align ring_hom.codomain_trivial_iff_range_trivial RingHom.codomain_trivial_iff_range_trivial /-- `f : α →+* β` doesn't map `1` to `0` if `β` is nontrivial -/ theorem map_one_ne_zero [Nontrivial β] : f 1 ≠ 0 := mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one #align ring_hom.map_one_ne_zero RingHom.map_one_ne_zero /-- If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. -/ theorem domain_nontrivial [Nontrivial β] : Nontrivial α := ⟨⟨1, 0, mt (fun h => show f 1 = 0 by rw [h, map_zero]) f.map_one_ne_zero⟩⟩ #align ring_hom.domain_nontrivial RingHom.domain_nontrivial theorem codomain_trivial (f : α →+* β) [h : Subsingleton α] : Subsingleton β := (subsingleton_or_nontrivial β).resolve_right fun _ => not_nontrivial_iff_subsingleton.mpr h f.domain_nontrivial #align ring_hom.codomain_trivial RingHom.codomain_trivial end /-- Ring homomorphisms preserve additive inverse. -/ protected theorem map_neg [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x : α) : f (-x) = -f x := map_neg f x #align ring_hom.map_neg RingHom.map_neg /-- Ring homomorphisms preserve subtraction. -/ protected theorem map_sub [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x y : α) : f (x - y) = f x - f y := map_sub f x y #align ring_hom.map_sub RingHom.map_sub /-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/ def mk' [NonAssocSemiring α] [NonAssocRing β] (f : α →* β) (map_add : ∀ a b, f (a + b) = f a + f b) : α →+* β := { AddMonoidHom.mk' f map_add, f with } #align ring_hom.mk' RingHom.mk' variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} /-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;> intros <;> rfl #align ring_hom.id RingHom.id instance : Inhabited (α →+* α) := ⟨id α⟩ @[simp] theorem id_apply (x : α) : RingHom.id α x = x := rfl #align ring_hom.id_apply RingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (id α : α →+ α) = AddMonoidHom.id α := rfl #align ring_hom.coe_add_monoid_hom_id RingHom.coe_addMonoidHom_id @[simp] theorem coe_monoidHom_id : (id α : α →* α) = MonoidHom.id α := rfl #align ring_hom.coe_monoid_hom_id RingHom.coe_monoidHom_id variable {_ : NonAssocSemiring γ} /-- Composition of ring homomorphisms is a ring homomorphism. -/ def comp (g : β →+* γ) (f : α →+* β) : α →+* γ := { g.toNonUnitalRingHom.comp f.toNonUnitalRingHom with toFun := g ∘ f, map_one' := by simp } #align ring_hom.comp RingHom.comp /-- Composition of semiring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonAssocSemiring δ} (f : α →+* β) (g : β →+* γ) (h : γ →+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align ring_hom.comp_assoc RingHom.comp_assoc @[simp] theorem coe_comp (hnp : β →+* γ) (hmn : α →+* β) : (hnp.comp hmn : α → γ) = hnp ∘ hmn := rfl #align ring_hom.coe_comp RingHom.coe_comp theorem comp_apply (hnp : β →+* γ) (hmn : α →+* β) (x : α) : (hnp.comp hmn : α → γ) x = hnp (hmn x) := rfl #align ring_hom.comp_apply RingHom.comp_apply @[simp] theorem comp_id (f : α →+* β) : f.comp (id α) = f := ext fun _ => rfl #align ring_hom.comp_id RingHom.comp_id @[simp] theorem id_comp (f : α →+* β) : (id β).comp f = f := ext fun _ => rfl #align ring_hom.id_comp RingHom.id_comp instance : Monoid (α →+* α) where one := id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ theorem one_def : (1 : α →+* α) = id α := rfl #align ring_hom.one_def RingHom.one_def theorem mul_def (f g : α →+* α) : f * g = f.comp g := rfl #align ring_hom.mul_def RingHom.mul_def @[simp] theorem coe_one : ⇑(1 : α →+* α) = _root_.id := rfl #align ring_hom.coe_one RingHom.coe_one @[simp] theorem coe_mul (f g : α →+* α) : ⇑(f * g) = f ∘ g := rfl #align ring_hom.coe_mul RingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →+* γ} {f : α →+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => RingHom.ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align ring_hom.cancel_right RingHom.cancel_right @[simp] theorem cancel_left {g : β →+* γ} {f₁ f₂ : α →+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => RingHom.ext fun x => hg <| by

rw [← comp_apply, h, comp_apply]

@[simp] theorem cancel_left {g : β →+* γ} {f₁ f₂ : α →+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => RingHom.ext fun x => hg <| by

Mathlib.Algebra.Ring.Hom.Defs.718_0.KyHvVYrIs9pW9ZQ

@[simp] theorem cancel_left {g : β →+* γ} {f₁ f₂ : α →+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂

Mathlib_Algebra_Ring_Hom_Defs

F : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : CommRing α inst✝¹ : IsDomain α inst✝ : CommRing β f : β →+ α h : ∀ (x : β), f (x * x) = f x * f x h_two : 2 ≠ 0 h_one : f 1 = 1 x y : β ⊢ OneHom.toFun { toFun := f.toFun, map_one' := h_one } (x * y) = OneHom.toFun { toFun := f.toFun, map_one' := h_one } x * OneHom.toFun { toFun := f.toFun, map_one' := h_one } y

/- Copyright (c) 2019 Amelia Livingston. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Amelia Livingston, Jireh Loreaux -/ import Mathlib.Algebra.Ring.Defs import Mathlib.Algebra.Ring.Basic import Mathlib.Data.Pi.Algebra #align_import algebra.hom.ring from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9" /-! # Homomorphisms of semirings and rings This file defines bundled homomorphisms of (non-unital) semirings and rings. As with monoid and groups, we use the same structure `RingHom a β`, a.k.a. `α →+* β`, for both types of homomorphisms. ## Main definitions * `NonUnitalRingHom`: Non-unital (semi)ring homomorphisms. Additive monoid homomorphism which preserve multiplication. * `RingHom`: (Semi)ring homomorphisms. Monoid homomorphisms which are also additive monoid homomorphism. ## Notations * `→ₙ+*`: Non-unital (semi)ring homs * `→+*`: (Semi)ring homs ## Implementation notes * There's a coercion from bundled homs to fun, and the canonical notation is to use the bundled hom as a function via this coercion. * There is no `SemiringHom` -- the idea is that `RingHom` is used. The constructor for a `RingHom` between semirings needs a proof of `map_zero`, `map_one` and `map_add` as well as `map_mul`; a separate constructor `RingHom.mk'` will construct ring homs between rings from monoid homs given only a proof that addition is preserved. ## Tags `RingHom`, `SemiringHom` -/ open Function variable {F α β γ : Type*} /-- Bundled non-unital semiring homomorphisms `α →ₙ+* β`; use this for bundled non-unital ring homomorphisms too. When possible, instead of parametrizing results over `(f : α →ₙ+* β)`, you should parametrize over `(F : Type*) [NonUnitalRingHomClass F α β] (f : F)`. When you extend this structure, make sure to extend `NonUnitalRingHomClass`. -/ structure NonUnitalRingHom (α β : Type*) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends α →ₙ* β, α →+ β #align non_unital_ring_hom NonUnitalRingHom /-- `α →ₙ+* β` denotes the type of non-unital ring homomorphisms from `α` to `β`. -/ infixr:25 " →ₙ+* " => NonUnitalRingHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as a semigroup homomorphism `α →ₙ* β`. The `simp`-normal form is `(f : α →ₙ* β)`. -/ add_decl_doc NonUnitalRingHom.toMulHom #align non_unital_ring_hom.to_mul_hom NonUnitalRingHom.toMulHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc NonUnitalRingHom.toAddMonoidHom #align non_unital_ring_hom.to_add_monoid_hom NonUnitalRingHom.toAddMonoidHom section NonUnitalRingHomClass /-- `NonUnitalRingHomClass F α β` states that `F` is a type of non-unital (semi)ring homomorphisms. You should extend this class when you extend `NonUnitalRingHom`. -/ class NonUnitalRingHomClass (F : Type*) (α β : outParam (Type*)) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends MulHomClass F α β, AddMonoidHomClass F α β #align non_unital_ring_hom_class NonUnitalRingHomClass variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] [NonUnitalRingHomClass F α β] /-- Turn an element of a type `F` satisfying `NonUnitalRingHomClass F α β` into an actual `NonUnitalRingHom`. This is declared as the default coercion from `F` to `α →ₙ+* β`. -/ @[coe] def NonUnitalRingHomClass.toNonUnitalRingHom (f : F) : α →ₙ+* β := { (f : α →ₙ* β), (f : α →+ β) with } /-- Any type satisfying `NonUnitalRingHomClass` can be cast into `NonUnitalRingHom` via `NonUnitalRingHomClass.toNonUnitalRingHom`. -/ instance : CoeTC F (α →ₙ+* β) := ⟨NonUnitalRingHomClass.toNonUnitalRingHom⟩ end NonUnitalRingHomClass namespace NonUnitalRingHom section coe variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] instance : NonUnitalRingHomClass (α →ₙ+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := NonUnitalRingHom.map_add' map_zero := NonUnitalRingHom.map_zero' map_mul f := f.map_mul' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -/ -- instance : CoeFun (α →ₙ+* β) fun _ => α → β := -- ⟨fun f => f.toFun⟩ -- Porting note: removed due to new `coe` in Lean4 #noalign non_unital_ring_hom.to_fun_eq_coe #noalign non_unital_ring_hom.coe_mk #noalign non_unital_ring_hom.coe_coe initialize_simps_projections NonUnitalRingHom (toFun → apply) @[simp] theorem coe_toMulHom (f : α →ₙ+* β) : ⇑f.toMulHom = f := rfl #align non_unital_ring_hom.coe_to_mul_hom NonUnitalRingHom.coe_toMulHom @[simp] theorem coe_mulHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →ₙ* β) = ⟨f, h₁⟩ := rfl #align non_unital_ring_hom.coe_mul_hom_mk NonUnitalRingHom.coe_mulHom_mk theorem coe_toAddMonoidHom (f : α →ₙ+* β) : ⇑f.toAddMonoidHom = f := rfl #align non_unital_ring_hom.coe_to_add_monoid_hom NonUnitalRingHom.coe_toAddMonoidHom @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →+ β) = ⟨⟨f, h₂⟩, h₃⟩ := rfl #align non_unital_ring_hom.coe_add_monoid_hom_mk NonUnitalRingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : α →ₙ+* β := { f.toMulHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align non_unital_ring_hom.copy NonUnitalRingHom.copy @[simp] theorem coe_copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align non_unital_ring_hom.coe_copy NonUnitalRingHom.coe_copy theorem copy_eq (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align non_unital_ring_hom.copy_eq NonUnitalRingHom.copy_eq end coe section variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] variable (f : α →ₙ+* β) {x y : α} @[ext] theorem ext ⦃f g : α →ₙ+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align non_unital_ring_hom.ext NonUnitalRingHom.ext theorem ext_iff {f g : α →ₙ+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align non_unital_ring_hom.ext_iff NonUnitalRingHom.ext_iff @[simp] theorem mk_coe (f : α →ₙ+* β) (h₁ h₂ h₃) : NonUnitalRingHom.mk (MulHom.mk f h₁) h₂ h₃ = f := ext fun _ => rfl #align non_unital_ring_hom.mk_coe NonUnitalRingHom.mk_coe theorem coe_addMonoidHom_injective : Injective fun f : α →ₙ+* β => (f : α →+ β) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align non_unital_ring_hom.coe_add_monoid_hom_injective NonUnitalRingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_mulHom_injective : Injective fun f : α →ₙ+* β => (f : α →ₙ* β) := fun _ _ h => ext <| MulHom.congr_fun h #align non_unital_ring_hom.coe_mul_hom_injective NonUnitalRingHom.coe_mulHom_injective end variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/ protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by refine' { toFun := id.. } <;> intros <;> rfl #align non_unital_ring_hom.id NonUnitalRingHom.id instance : Zero (α →ₙ+* β) := ⟨{ toFun := 0, map_mul' := fun _ _ => (mul_zero (0 : β)).symm, map_zero' := rfl, map_add' := fun _ _ => (add_zero (0 : β)).symm }⟩ instance : Inhabited (α →ₙ+* β) := ⟨0⟩ @[simp] theorem coe_zero : ⇑(0 : α →ₙ+* β) = 0 := rfl #align non_unital_ring_hom.coe_zero NonUnitalRingHom.coe_zero @[simp] theorem zero_apply (x : α) : (0 : α →ₙ+* β) x = 0 := rfl #align non_unital_ring_hom.zero_apply NonUnitalRingHom.zero_apply @[simp] theorem id_apply (x : α) : NonUnitalRingHom.id α x = x := rfl #align non_unital_ring_hom.id_apply NonUnitalRingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (NonUnitalRingHom.id α : α →+ α) = AddMonoidHom.id α := rfl #align non_unital_ring_hom.coe_add_monoid_hom_id NonUnitalRingHom.coe_addMonoidHom_id @[simp] theorem coe_mulHom_id : (NonUnitalRingHom.id α : α →ₙ* α) = MulHom.id α := rfl #align non_unital_ring_hom.coe_mul_hom_id NonUnitalRingHom.coe_mulHom_id variable [NonUnitalNonAssocSemiring γ] /-- Composition of non-unital ring homomorphisms is a non-unital ring homomorphism. -/ def comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : α →ₙ+* γ := { g.toMulHom.comp f.toMulHom, g.toAddMonoidHom.comp f.toAddMonoidHom with } #align non_unital_ring_hom.comp NonUnitalRingHom.comp /-- Composition of non-unital ring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonUnitalNonAssocSemiring δ} (f : α →ₙ+* β) (g : β →ₙ+* γ) (h : γ →ₙ+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align non_unital_ring_hom.comp_assoc NonUnitalRingHom.comp_assoc @[simp] theorem coe_comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : ⇑(g.comp f) = g ∘ f := rfl #align non_unital_ring_hom.coe_comp NonUnitalRingHom.coe_comp @[simp] theorem comp_apply (g : β →ₙ+* γ) (f : α →ₙ+* β) (x : α) : g.comp f x = g (f x) := rfl #align non_unital_ring_hom.comp_apply NonUnitalRingHom.comp_apply variable (g : β →ₙ+* γ) (f : α →ₙ+* β) @[simp] theorem coe_comp_addMonoidHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : AddMonoidHom.mk ⟨g ∘ f, (g.comp f).map_zero'⟩ (g.comp f).map_add' = (g : β →+ γ).comp f := rfl #align non_unital_ring_hom.coe_comp_add_monoid_hom NonUnitalRingHom.coe_comp_addMonoidHom @[simp] theorem coe_comp_mulHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : MulHom.mk (g ∘ f) (g.comp f).map_mul' = (g : β →ₙ* γ).comp f := rfl #align non_unital_ring_hom.coe_comp_mul_hom NonUnitalRingHom.coe_comp_mulHom @[simp] theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 := by ext simp #align non_unital_ring_hom.comp_zero NonUnitalRingHom.comp_zero @[simp] theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 := by ext rfl #align non_unital_ring_hom.zero_comp NonUnitalRingHom.zero_comp @[simp] theorem comp_id (f : α →ₙ+* β) : f.comp (NonUnitalRingHom.id α) = f := ext fun _ => rfl #align non_unital_ring_hom.comp_id NonUnitalRingHom.comp_id @[simp] theorem id_comp (f : α →ₙ+* β) : (NonUnitalRingHom.id β).comp f = f := ext fun _ => rfl #align non_unital_ring_hom.id_comp NonUnitalRingHom.id_comp instance : MonoidWithZero (α →ₙ+* α) where one := NonUnitalRingHom.id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ zero := 0 mul_zero := comp_zero zero_mul := zero_comp theorem one_def : (1 : α →ₙ+* α) = NonUnitalRingHom.id α := rfl #align non_unital_ring_hom.one_def NonUnitalRingHom.one_def @[simp] theorem coe_one : ⇑(1 : α →ₙ+* α) = id := rfl #align non_unital_ring_hom.coe_one NonUnitalRingHom.coe_one theorem mul_def (f g : α →ₙ+* α) : f * g = f.comp g := rfl #align non_unital_ring_hom.mul_def NonUnitalRingHom.mul_def @[simp] theorem coe_mul (f g : α →ₙ+* α) : ⇑(f * g) = f ∘ g := rfl #align non_unital_ring_hom.coe_mul NonUnitalRingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →ₙ+* γ} {f : α →ₙ+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_right NonUnitalRingHom.cancel_right @[simp] theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_left NonUnitalRingHom.cancel_left end NonUnitalRingHom /-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too. This extends from both `MonoidHom` and `MonoidWithZeroHom` in order to put the fields in a sensible order, even though `MonoidWithZeroHom` already extends `MonoidHom`. -/ structure RingHom (α : Type*) (β : Type*) [NonAssocSemiring α] [NonAssocSemiring β] extends α →* β, α →+ β, α →ₙ+* β, α →*₀ β #align ring_hom RingHom /-- `α →+* β` denotes the type of ring homomorphisms from `α` to `β`. -/ infixr:25 " →+* " => RingHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid with zero homomorphism `α →*₀ β`. The `simp`-normal form is `(f : α →*₀ β)`. -/ add_decl_doc RingHom.toMonoidWithZeroHom #align ring_hom.to_monoid_with_zero_hom RingHom.toMonoidWithZeroHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid homomorphism `α →* β`. The `simp`-normal form is `(f : α →* β)`. -/ add_decl_doc RingHom.toMonoidHom #align ring_hom.to_monoid_hom RingHom.toMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc RingHom.toAddMonoidHom #align ring_hom.to_add_monoid_hom RingHom.toAddMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a non-unital ring homomorphism `α →ₙ+* β`. The `simp`-normal form is `(f : α →ₙ+* β)`. -/ add_decl_doc RingHom.toNonUnitalRingHom #align ring_hom.to_non_unital_ring_hom RingHom.toNonUnitalRingHom section RingHomClass /-- `RingHomClass F α β` states that `F` is a type of (semi)ring homomorphisms. You should extend this class when you extend `RingHom`. This extends from both `MonoidHomClass` and `MonoidWithZeroHomClass` in order to put the fields in a sensible order, even though `MonoidWithZeroHomClass` already extends `MonoidHomClass`. -/ class RingHomClass (F : Type*) (α β : outParam (Type*)) [NonAssocSemiring α] [NonAssocSemiring β] extends MonoidHomClass F α β, AddMonoidHomClass F α β, MonoidWithZeroHomClass F α β #align ring_hom_class RingHomClass set_option linter.deprecated false in /-- Ring homomorphisms preserve `bit1`. -/ @[simp] lemma map_bit1 [NonAssocSemiring α] [NonAssocSemiring β] [RingHomClass F α β] (f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) := by simp [bit1] #align map_bit1 map_bit1 -- Porting note: marked `{}` rather than `[]` to prevent dangerous instances variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} [RingHomClass F α β] /-- Turn an element of a type `F` satisfying `RingHomClass F α β` into an actual `RingHom`. This is declared as the default coercion from `F` to `α →+* β`. -/ @[coe] def RingHomClass.toRingHom (f : F) : α →+* β := { (f : α →* β), (f : α →+ β) with } /-- Any type satisfying `RingHomClass` can be cast into `RingHom` via `RingHomClass.toRingHom`. -/ instance : CoeTC F (α →+* β) := ⟨RingHomClass.toRingHom⟩ instance (priority := 100) RingHomClass.toNonUnitalRingHomClass : NonUnitalRingHomClass F α β := { ‹RingHomClass F α β› with } #align ring_hom_class.to_non_unital_ring_hom_class RingHomClass.toNonUnitalRingHomClass end RingHomClass namespace RingHom section coe /-! Throughout this section, some `Semiring` arguments are specified with `{}` instead of `[]`. See note [implicit instance arguments]. -/ variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} instance instRingHomClass : RingHomClass (α →+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := RingHom.map_add' map_zero := RingHom.map_zero' map_mul f := f.map_mul' map_one f := f.map_one' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -- -/ -- instance : CoeFun (α →+* β) fun _ => α → β := -- ⟨RingHom.toFun⟩ initialize_simps_projections RingHom (toFun → apply) -- Porting note: is this lemma still needed in Lean4? -- Porting note: because `f.toFun` really means `f.toMonoidHom.toOneHom.toFun` and -- `toMonoidHom_eq_coe` wants to simplify `f.toMonoidHom` to `(↑f : M →* N)`, this can't -- be a simp lemma anymore -- @[simp] theorem toFun_eq_coe (f : α →+* β) : f.toFun = f := rfl #align ring_hom.to_fun_eq_coe RingHom.toFun_eq_coe @[simp] theorem coe_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α → β) = f := rfl #align ring_hom.coe_mk RingHom.coe_mk @[simp] theorem coe_coe {F : Type*} [RingHomClass F α β] (f : F) : ((f : α →+* β) : α → β) = f := rfl #align ring_hom.coe_coe RingHom.coe_coe attribute [coe] RingHom.toMonoidHom instance coeToMonoidHom : Coe (α →+* β) (α →* β) := ⟨RingHom.toMonoidHom⟩ #align ring_hom.has_coe_monoid_hom RingHom.coeToMonoidHom -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_monoid_hom @[simp] theorem toMonoidHom_eq_coe (f : α →+* β) : f.toMonoidHom = f := rfl #align ring_hom.to_monoid_hom_eq_coe RingHom.toMonoidHom_eq_coe -- Porting note: this can't be a simp lemma anymore -- @[simp] theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f := by rfl #align ring_hom.to_monoid_with_zero_hom_eq_coe RingHom.toMonoidWithZeroHom_eq_coe @[simp] theorem coe_monoidHom_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α →* β) = f := rfl #align ring_hom.coe_monoid_hom_mk RingHom.coe_monoidHom_mk -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_add_monoid_hom @[simp] theorem toAddMonoidHom_eq_coe (f : α →+* β) : f.toAddMonoidHom = f := rfl #align ring_hom.to_add_monoid_hom_eq_coe RingHom.toAddMonoidHom_eq_coe @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃ h₄) : ((⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨⟨f, h₃⟩, h₄⟩ := rfl #align ring_hom.coe_add_monoid_hom_mk RingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ def copy (f : α →+* β) (f' : α → β) (h : f' = f) : α →+* β := { f.toMonoidWithZeroHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align ring_hom.copy RingHom.copy @[simp] theorem coe_copy (f : α →+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align ring_hom.coe_copy RingHom.coe_copy theorem copy_eq (f : α →+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align ring_hom.copy_eq RingHom.copy_eq end coe section variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} (f : α →+* β) {x y : α} theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x := FunLike.congr_fun h x #align ring_hom.congr_fun RingHom.congr_fun theorem congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y := FunLike.congr_arg f h #align ring_hom.congr_arg RingHom.congr_arg theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g := FunLike.coe_injective h #align ring_hom.coe_inj RingHom.coe_inj @[ext] theorem ext ⦃f g : α →+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align ring_hom.ext RingHom.ext theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align ring_hom.ext_iff RingHom.ext_iff @[simp] theorem mk_coe (f : α →+* β) (h₁ h₂ h₃ h₄) : RingHom.mk ⟨⟨f, h₁⟩, h₂⟩ h₃ h₄ = f := ext fun _ => rfl #align ring_hom.mk_coe RingHom.mk_coe theorem coe_addMonoidHom_injective : Injective (fun f : α →+* β => (f : α →+ β)) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align ring_hom.coe_add_monoid_hom_injective RingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_monoidHom_injective : Injective (fun f : α →+* β => (f : α →* β)) := fun _ _ h => ext <| MonoidHom.congr_fun h #align ring_hom.coe_monoid_hom_injective RingHom.coe_monoidHom_injective /-- Ring homomorphisms map zero to zero. -/ protected theorem map_zero (f : α →+* β) : f 0 = 0 := map_zero f #align ring_hom.map_zero RingHom.map_zero /-- Ring homomorphisms map one to one. -/ protected theorem map_one (f : α →+* β) : f 1 = 1 := map_one f #align ring_hom.map_one RingHom.map_one /-- Ring homomorphisms preserve addition. -/ protected theorem map_add (f : α →+* β) : ∀ a b, f (a + b) = f a + f b := map_add f #align ring_hom.map_add RingHom.map_add /-- Ring homomorphisms preserve multiplication. -/ protected theorem map_mul (f : α →+* β) : ∀ a b, f (a * b) = f a * f b := map_mul f #align ring_hom.map_mul RingHom.map_mul @[simp] theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 0 1) = ite p 0 1 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_zero_one RingHom.map_ite_zero_one @[simp] theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 1 0) = ite p 1 0 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_one_zero RingHom.map_ite_one_zero /-- `f : α →+* β` has a trivial codomain iff `f 1 = 0`. -/ theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 := by rw [map_one, eq_comm] #align ring_hom.codomain_trivial_iff_map_one_eq_zero RingHom.codomain_trivial_iff_map_one_eq_zero /-- `f : α →+* β` has a trivial codomain iff it has a trivial range. -/ theorem codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 := f.codomain_trivial_iff_map_one_eq_zero.trans ⟨fun h x => by rw [← mul_one x, map_mul, h, mul_zero], fun h => h 1⟩ #align ring_hom.codomain_trivial_iff_range_trivial RingHom.codomain_trivial_iff_range_trivial /-- `f : α →+* β` doesn't map `1` to `0` if `β` is nontrivial -/ theorem map_one_ne_zero [Nontrivial β] : f 1 ≠ 0 := mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one #align ring_hom.map_one_ne_zero RingHom.map_one_ne_zero /-- If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. -/ theorem domain_nontrivial [Nontrivial β] : Nontrivial α := ⟨⟨1, 0, mt (fun h => show f 1 = 0 by rw [h, map_zero]) f.map_one_ne_zero⟩⟩ #align ring_hom.domain_nontrivial RingHom.domain_nontrivial theorem codomain_trivial (f : α →+* β) [h : Subsingleton α] : Subsingleton β := (subsingleton_or_nontrivial β).resolve_right fun _ => not_nontrivial_iff_subsingleton.mpr h f.domain_nontrivial #align ring_hom.codomain_trivial RingHom.codomain_trivial end /-- Ring homomorphisms preserve additive inverse. -/ protected theorem map_neg [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x : α) : f (-x) = -f x := map_neg f x #align ring_hom.map_neg RingHom.map_neg /-- Ring homomorphisms preserve subtraction. -/ protected theorem map_sub [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x y : α) : f (x - y) = f x - f y := map_sub f x y #align ring_hom.map_sub RingHom.map_sub /-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/ def mk' [NonAssocSemiring α] [NonAssocRing β] (f : α →* β) (map_add : ∀ a b, f (a + b) = f a + f b) : α →+* β := { AddMonoidHom.mk' f map_add, f with } #align ring_hom.mk' RingHom.mk' variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} /-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;> intros <;> rfl #align ring_hom.id RingHom.id instance : Inhabited (α →+* α) := ⟨id α⟩ @[simp] theorem id_apply (x : α) : RingHom.id α x = x := rfl #align ring_hom.id_apply RingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (id α : α →+ α) = AddMonoidHom.id α := rfl #align ring_hom.coe_add_monoid_hom_id RingHom.coe_addMonoidHom_id @[simp] theorem coe_monoidHom_id : (id α : α →* α) = MonoidHom.id α := rfl #align ring_hom.coe_monoid_hom_id RingHom.coe_monoidHom_id variable {_ : NonAssocSemiring γ} /-- Composition of ring homomorphisms is a ring homomorphism. -/ def comp (g : β →+* γ) (f : α →+* β) : α →+* γ := { g.toNonUnitalRingHom.comp f.toNonUnitalRingHom with toFun := g ∘ f, map_one' := by simp } #align ring_hom.comp RingHom.comp /-- Composition of semiring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonAssocSemiring δ} (f : α →+* β) (g : β →+* γ) (h : γ →+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align ring_hom.comp_assoc RingHom.comp_assoc @[simp] theorem coe_comp (hnp : β →+* γ) (hmn : α →+* β) : (hnp.comp hmn : α → γ) = hnp ∘ hmn := rfl #align ring_hom.coe_comp RingHom.coe_comp theorem comp_apply (hnp : β →+* γ) (hmn : α →+* β) (x : α) : (hnp.comp hmn : α → γ) x = hnp (hmn x) := rfl #align ring_hom.comp_apply RingHom.comp_apply @[simp] theorem comp_id (f : α →+* β) : f.comp (id α) = f := ext fun _ => rfl #align ring_hom.comp_id RingHom.comp_id @[simp] theorem id_comp (f : α →+* β) : (id β).comp f = f := ext fun _ => rfl #align ring_hom.id_comp RingHom.id_comp instance : Monoid (α →+* α) where one := id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ theorem one_def : (1 : α →+* α) = id α := rfl #align ring_hom.one_def RingHom.one_def theorem mul_def (f g : α →+* α) : f * g = f.comp g := rfl #align ring_hom.mul_def RingHom.mul_def @[simp] theorem coe_one : ⇑(1 : α →+* α) = _root_.id := rfl #align ring_hom.coe_one RingHom.coe_one @[simp] theorem coe_mul (f g : α →+* α) : ⇑(f * g) = f ∘ g := rfl #align ring_hom.coe_mul RingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →+* γ} {f : α →+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => RingHom.ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align ring_hom.cancel_right RingHom.cancel_right @[simp] theorem cancel_left {g : β →+* γ} {f₁ f₂ : α →+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => RingHom.ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align ring_hom.cancel_left RingHom.cancel_left end RingHom namespace AddMonoidHom variable [CommRing α] [IsDomain α] [CommRing β] (f : β →+ α) -- Porting note: there's some disagreement over the naming scheme here. -- This could perhaps be `mkRingHom_of_mul_self_of_two_ne_zero`. -- See https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/naming.20conventions/near/315558410 /-- Make a ring homomorphism from an additive group homomorphism from a commutative ring to an integral domain that commutes with self multiplication, assumes that two is nonzero and `1` is sent to `1`. -/ def mkRingHomOfMulSelfOfTwoNeZero (h : ∀ x, f (x * x) = f x * f x) (h_two : (2 : α) ≠ 0) (h_one : f 1 = 1) : β →+* α := { f with map_one' := h_one, map_mul' := fun x y => by

have hxy := h (x + y)

/-- Make a ring homomorphism from an additive group homomorphism from a commutative ring to an integral domain that commutes with self multiplication, assumes that two is nonzero and `1` is sent to `1`. -/ def mkRingHomOfMulSelfOfTwoNeZero (h : ∀ x, f (x * x) = f x * f x) (h_two : (2 : α) ≠ 0) (h_one : f 1 = 1) : β →+* α := { f with map_one' := h_one, map_mul' := fun x y => by

Mathlib.Algebra.Ring.Hom.Defs.733_0.KyHvVYrIs9pW9ZQ

/-- Make a ring homomorphism from an additive group homomorphism from a commutative ring to an integral domain that commutes with self multiplication, assumes that two is nonzero and `1` is sent to `1`. -/ def mkRingHomOfMulSelfOfTwoNeZero (h : ∀ x, f (x * x) = f x * f x) (h_two : (2 : α) ≠ 0) (h_one : f 1 = 1) : β →+* α

Mathlib_Algebra_Ring_Hom_Defs

F : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : CommRing α inst✝¹ : IsDomain α inst✝ : CommRing β f : β →+ α h : ∀ (x : β), f (x * x) = f x * f x h_two : 2 ≠ 0 h_one : f 1 = 1 x y : β hxy : f ((x + y) * (x + y)) = f (x + y) * f (x + y) ⊢ OneHom.toFun { toFun := f.toFun, map_one' := h_one } (x * y) = OneHom.toFun { toFun := f.toFun, map_one' := h_one } x * OneHom.toFun { toFun := f.toFun, map_one' := h_one } y

/- Copyright (c) 2019 Amelia Livingston. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Amelia Livingston, Jireh Loreaux -/ import Mathlib.Algebra.Ring.Defs import Mathlib.Algebra.Ring.Basic import Mathlib.Data.Pi.Algebra #align_import algebra.hom.ring from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9" /-! # Homomorphisms of semirings and rings This file defines bundled homomorphisms of (non-unital) semirings and rings. As with monoid and groups, we use the same structure `RingHom a β`, a.k.a. `α →+* β`, for both types of homomorphisms. ## Main definitions * `NonUnitalRingHom`: Non-unital (semi)ring homomorphisms. Additive monoid homomorphism which preserve multiplication. * `RingHom`: (Semi)ring homomorphisms. Monoid homomorphisms which are also additive monoid homomorphism. ## Notations * `→ₙ+*`: Non-unital (semi)ring homs * `→+*`: (Semi)ring homs ## Implementation notes * There's a coercion from bundled homs to fun, and the canonical notation is to use the bundled hom as a function via this coercion. * There is no `SemiringHom` -- the idea is that `RingHom` is used. The constructor for a `RingHom` between semirings needs a proof of `map_zero`, `map_one` and `map_add` as well as `map_mul`; a separate constructor `RingHom.mk'` will construct ring homs between rings from monoid homs given only a proof that addition is preserved. ## Tags `RingHom`, `SemiringHom` -/ open Function variable {F α β γ : Type*} /-- Bundled non-unital semiring homomorphisms `α →ₙ+* β`; use this for bundled non-unital ring homomorphisms too. When possible, instead of parametrizing results over `(f : α →ₙ+* β)`, you should parametrize over `(F : Type*) [NonUnitalRingHomClass F α β] (f : F)`. When you extend this structure, make sure to extend `NonUnitalRingHomClass`. -/ structure NonUnitalRingHom (α β : Type*) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends α →ₙ* β, α →+ β #align non_unital_ring_hom NonUnitalRingHom /-- `α →ₙ+* β` denotes the type of non-unital ring homomorphisms from `α` to `β`. -/ infixr:25 " →ₙ+* " => NonUnitalRingHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as a semigroup homomorphism `α →ₙ* β`. The `simp`-normal form is `(f : α →ₙ* β)`. -/ add_decl_doc NonUnitalRingHom.toMulHom #align non_unital_ring_hom.to_mul_hom NonUnitalRingHom.toMulHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc NonUnitalRingHom.toAddMonoidHom #align non_unital_ring_hom.to_add_monoid_hom NonUnitalRingHom.toAddMonoidHom section NonUnitalRingHomClass /-- `NonUnitalRingHomClass F α β` states that `F` is a type of non-unital (semi)ring homomorphisms. You should extend this class when you extend `NonUnitalRingHom`. -/ class NonUnitalRingHomClass (F : Type*) (α β : outParam (Type*)) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends MulHomClass F α β, AddMonoidHomClass F α β #align non_unital_ring_hom_class NonUnitalRingHomClass variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] [NonUnitalRingHomClass F α β] /-- Turn an element of a type `F` satisfying `NonUnitalRingHomClass F α β` into an actual `NonUnitalRingHom`. This is declared as the default coercion from `F` to `α →ₙ+* β`. -/ @[coe] def NonUnitalRingHomClass.toNonUnitalRingHom (f : F) : α →ₙ+* β := { (f : α →ₙ* β), (f : α →+ β) with } /-- Any type satisfying `NonUnitalRingHomClass` can be cast into `NonUnitalRingHom` via `NonUnitalRingHomClass.toNonUnitalRingHom`. -/ instance : CoeTC F (α →ₙ+* β) := ⟨NonUnitalRingHomClass.toNonUnitalRingHom⟩ end NonUnitalRingHomClass namespace NonUnitalRingHom section coe variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] instance : NonUnitalRingHomClass (α →ₙ+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := NonUnitalRingHom.map_add' map_zero := NonUnitalRingHom.map_zero' map_mul f := f.map_mul' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -/ -- instance : CoeFun (α →ₙ+* β) fun _ => α → β := -- ⟨fun f => f.toFun⟩ -- Porting note: removed due to new `coe` in Lean4 #noalign non_unital_ring_hom.to_fun_eq_coe #noalign non_unital_ring_hom.coe_mk #noalign non_unital_ring_hom.coe_coe initialize_simps_projections NonUnitalRingHom (toFun → apply) @[simp] theorem coe_toMulHom (f : α →ₙ+* β) : ⇑f.toMulHom = f := rfl #align non_unital_ring_hom.coe_to_mul_hom NonUnitalRingHom.coe_toMulHom @[simp] theorem coe_mulHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →ₙ* β) = ⟨f, h₁⟩ := rfl #align non_unital_ring_hom.coe_mul_hom_mk NonUnitalRingHom.coe_mulHom_mk theorem coe_toAddMonoidHom (f : α →ₙ+* β) : ⇑f.toAddMonoidHom = f := rfl #align non_unital_ring_hom.coe_to_add_monoid_hom NonUnitalRingHom.coe_toAddMonoidHom @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →+ β) = ⟨⟨f, h₂⟩, h₃⟩ := rfl #align non_unital_ring_hom.coe_add_monoid_hom_mk NonUnitalRingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : α →ₙ+* β := { f.toMulHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align non_unital_ring_hom.copy NonUnitalRingHom.copy @[simp] theorem coe_copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align non_unital_ring_hom.coe_copy NonUnitalRingHom.coe_copy theorem copy_eq (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align non_unital_ring_hom.copy_eq NonUnitalRingHom.copy_eq end coe section variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] variable (f : α →ₙ+* β) {x y : α} @[ext] theorem ext ⦃f g : α →ₙ+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align non_unital_ring_hom.ext NonUnitalRingHom.ext theorem ext_iff {f g : α →ₙ+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align non_unital_ring_hom.ext_iff NonUnitalRingHom.ext_iff @[simp] theorem mk_coe (f : α →ₙ+* β) (h₁ h₂ h₃) : NonUnitalRingHom.mk (MulHom.mk f h₁) h₂ h₃ = f := ext fun _ => rfl #align non_unital_ring_hom.mk_coe NonUnitalRingHom.mk_coe theorem coe_addMonoidHom_injective : Injective fun f : α →ₙ+* β => (f : α →+ β) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align non_unital_ring_hom.coe_add_monoid_hom_injective NonUnitalRingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_mulHom_injective : Injective fun f : α →ₙ+* β => (f : α →ₙ* β) := fun _ _ h => ext <| MulHom.congr_fun h #align non_unital_ring_hom.coe_mul_hom_injective NonUnitalRingHom.coe_mulHom_injective end variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/ protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by refine' { toFun := id.. } <;> intros <;> rfl #align non_unital_ring_hom.id NonUnitalRingHom.id instance : Zero (α →ₙ+* β) := ⟨{ toFun := 0, map_mul' := fun _ _ => (mul_zero (0 : β)).symm, map_zero' := rfl, map_add' := fun _ _ => (add_zero (0 : β)).symm }⟩ instance : Inhabited (α →ₙ+* β) := ⟨0⟩ @[simp] theorem coe_zero : ⇑(0 : α →ₙ+* β) = 0 := rfl #align non_unital_ring_hom.coe_zero NonUnitalRingHom.coe_zero @[simp] theorem zero_apply (x : α) : (0 : α →ₙ+* β) x = 0 := rfl #align non_unital_ring_hom.zero_apply NonUnitalRingHom.zero_apply @[simp] theorem id_apply (x : α) : NonUnitalRingHom.id α x = x := rfl #align non_unital_ring_hom.id_apply NonUnitalRingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (NonUnitalRingHom.id α : α →+ α) = AddMonoidHom.id α := rfl #align non_unital_ring_hom.coe_add_monoid_hom_id NonUnitalRingHom.coe_addMonoidHom_id @[simp] theorem coe_mulHom_id : (NonUnitalRingHom.id α : α →ₙ* α) = MulHom.id α := rfl #align non_unital_ring_hom.coe_mul_hom_id NonUnitalRingHom.coe_mulHom_id variable [NonUnitalNonAssocSemiring γ] /-- Composition of non-unital ring homomorphisms is a non-unital ring homomorphism. -/ def comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : α →ₙ+* γ := { g.toMulHom.comp f.toMulHom, g.toAddMonoidHom.comp f.toAddMonoidHom with } #align non_unital_ring_hom.comp NonUnitalRingHom.comp /-- Composition of non-unital ring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonUnitalNonAssocSemiring δ} (f : α →ₙ+* β) (g : β →ₙ+* γ) (h : γ →ₙ+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align non_unital_ring_hom.comp_assoc NonUnitalRingHom.comp_assoc @[simp] theorem coe_comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : ⇑(g.comp f) = g ∘ f := rfl #align non_unital_ring_hom.coe_comp NonUnitalRingHom.coe_comp @[simp] theorem comp_apply (g : β →ₙ+* γ) (f : α →ₙ+* β) (x : α) : g.comp f x = g (f x) := rfl #align non_unital_ring_hom.comp_apply NonUnitalRingHom.comp_apply variable (g : β →ₙ+* γ) (f : α →ₙ+* β) @[simp] theorem coe_comp_addMonoidHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : AddMonoidHom.mk ⟨g ∘ f, (g.comp f).map_zero'⟩ (g.comp f).map_add' = (g : β →+ γ).comp f := rfl #align non_unital_ring_hom.coe_comp_add_monoid_hom NonUnitalRingHom.coe_comp_addMonoidHom @[simp] theorem coe_comp_mulHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : MulHom.mk (g ∘ f) (g.comp f).map_mul' = (g : β →ₙ* γ).comp f := rfl #align non_unital_ring_hom.coe_comp_mul_hom NonUnitalRingHom.coe_comp_mulHom @[simp] theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 := by ext simp #align non_unital_ring_hom.comp_zero NonUnitalRingHom.comp_zero @[simp] theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 := by ext rfl #align non_unital_ring_hom.zero_comp NonUnitalRingHom.zero_comp @[simp] theorem comp_id (f : α →ₙ+* β) : f.comp (NonUnitalRingHom.id α) = f := ext fun _ => rfl #align non_unital_ring_hom.comp_id NonUnitalRingHom.comp_id @[simp] theorem id_comp (f : α →ₙ+* β) : (NonUnitalRingHom.id β).comp f = f := ext fun _ => rfl #align non_unital_ring_hom.id_comp NonUnitalRingHom.id_comp instance : MonoidWithZero (α →ₙ+* α) where one := NonUnitalRingHom.id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ zero := 0 mul_zero := comp_zero zero_mul := zero_comp theorem one_def : (1 : α →ₙ+* α) = NonUnitalRingHom.id α := rfl #align non_unital_ring_hom.one_def NonUnitalRingHom.one_def @[simp] theorem coe_one : ⇑(1 : α →ₙ+* α) = id := rfl #align non_unital_ring_hom.coe_one NonUnitalRingHom.coe_one theorem mul_def (f g : α →ₙ+* α) : f * g = f.comp g := rfl #align non_unital_ring_hom.mul_def NonUnitalRingHom.mul_def @[simp] theorem coe_mul (f g : α →ₙ+* α) : ⇑(f * g) = f ∘ g := rfl #align non_unital_ring_hom.coe_mul NonUnitalRingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →ₙ+* γ} {f : α →ₙ+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_right NonUnitalRingHom.cancel_right @[simp] theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_left NonUnitalRingHom.cancel_left end NonUnitalRingHom /-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too. This extends from both `MonoidHom` and `MonoidWithZeroHom` in order to put the fields in a sensible order, even though `MonoidWithZeroHom` already extends `MonoidHom`. -/ structure RingHom (α : Type*) (β : Type*) [NonAssocSemiring α] [NonAssocSemiring β] extends α →* β, α →+ β, α →ₙ+* β, α →*₀ β #align ring_hom RingHom /-- `α →+* β` denotes the type of ring homomorphisms from `α` to `β`. -/ infixr:25 " →+* " => RingHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid with zero homomorphism `α →*₀ β`. The `simp`-normal form is `(f : α →*₀ β)`. -/ add_decl_doc RingHom.toMonoidWithZeroHom #align ring_hom.to_monoid_with_zero_hom RingHom.toMonoidWithZeroHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid homomorphism `α →* β`. The `simp`-normal form is `(f : α →* β)`. -/ add_decl_doc RingHom.toMonoidHom #align ring_hom.to_monoid_hom RingHom.toMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc RingHom.toAddMonoidHom #align ring_hom.to_add_monoid_hom RingHom.toAddMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a non-unital ring homomorphism `α →ₙ+* β`. The `simp`-normal form is `(f : α →ₙ+* β)`. -/ add_decl_doc RingHom.toNonUnitalRingHom #align ring_hom.to_non_unital_ring_hom RingHom.toNonUnitalRingHom section RingHomClass /-- `RingHomClass F α β` states that `F` is a type of (semi)ring homomorphisms. You should extend this class when you extend `RingHom`. This extends from both `MonoidHomClass` and `MonoidWithZeroHomClass` in order to put the fields in a sensible order, even though `MonoidWithZeroHomClass` already extends `MonoidHomClass`. -/ class RingHomClass (F : Type*) (α β : outParam (Type*)) [NonAssocSemiring α] [NonAssocSemiring β] extends MonoidHomClass F α β, AddMonoidHomClass F α β, MonoidWithZeroHomClass F α β #align ring_hom_class RingHomClass set_option linter.deprecated false in /-- Ring homomorphisms preserve `bit1`. -/ @[simp] lemma map_bit1 [NonAssocSemiring α] [NonAssocSemiring β] [RingHomClass F α β] (f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) := by simp [bit1] #align map_bit1 map_bit1 -- Porting note: marked `{}` rather than `[]` to prevent dangerous instances variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} [RingHomClass F α β] /-- Turn an element of a type `F` satisfying `RingHomClass F α β` into an actual `RingHom`. This is declared as the default coercion from `F` to `α →+* β`. -/ @[coe] def RingHomClass.toRingHom (f : F) : α →+* β := { (f : α →* β), (f : α →+ β) with } /-- Any type satisfying `RingHomClass` can be cast into `RingHom` via `RingHomClass.toRingHom`. -/ instance : CoeTC F (α →+* β) := ⟨RingHomClass.toRingHom⟩ instance (priority := 100) RingHomClass.toNonUnitalRingHomClass : NonUnitalRingHomClass F α β := { ‹RingHomClass F α β› with } #align ring_hom_class.to_non_unital_ring_hom_class RingHomClass.toNonUnitalRingHomClass end RingHomClass namespace RingHom section coe /-! Throughout this section, some `Semiring` arguments are specified with `{}` instead of `[]`. See note [implicit instance arguments]. -/ variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} instance instRingHomClass : RingHomClass (α →+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := RingHom.map_add' map_zero := RingHom.map_zero' map_mul f := f.map_mul' map_one f := f.map_one' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -- -/ -- instance : CoeFun (α →+* β) fun _ => α → β := -- ⟨RingHom.toFun⟩ initialize_simps_projections RingHom (toFun → apply) -- Porting note: is this lemma still needed in Lean4? -- Porting note: because `f.toFun` really means `f.toMonoidHom.toOneHom.toFun` and -- `toMonoidHom_eq_coe` wants to simplify `f.toMonoidHom` to `(↑f : M →* N)`, this can't -- be a simp lemma anymore -- @[simp] theorem toFun_eq_coe (f : α →+* β) : f.toFun = f := rfl #align ring_hom.to_fun_eq_coe RingHom.toFun_eq_coe @[simp] theorem coe_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α → β) = f := rfl #align ring_hom.coe_mk RingHom.coe_mk @[simp] theorem coe_coe {F : Type*} [RingHomClass F α β] (f : F) : ((f : α →+* β) : α → β) = f := rfl #align ring_hom.coe_coe RingHom.coe_coe attribute [coe] RingHom.toMonoidHom instance coeToMonoidHom : Coe (α →+* β) (α →* β) := ⟨RingHom.toMonoidHom⟩ #align ring_hom.has_coe_monoid_hom RingHom.coeToMonoidHom -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_monoid_hom @[simp] theorem toMonoidHom_eq_coe (f : α →+* β) : f.toMonoidHom = f := rfl #align ring_hom.to_monoid_hom_eq_coe RingHom.toMonoidHom_eq_coe -- Porting note: this can't be a simp lemma anymore -- @[simp] theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f := by rfl #align ring_hom.to_monoid_with_zero_hom_eq_coe RingHom.toMonoidWithZeroHom_eq_coe @[simp] theorem coe_monoidHom_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α →* β) = f := rfl #align ring_hom.coe_monoid_hom_mk RingHom.coe_monoidHom_mk -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_add_monoid_hom @[simp] theorem toAddMonoidHom_eq_coe (f : α →+* β) : f.toAddMonoidHom = f := rfl #align ring_hom.to_add_monoid_hom_eq_coe RingHom.toAddMonoidHom_eq_coe @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃ h₄) : ((⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨⟨f, h₃⟩, h₄⟩ := rfl #align ring_hom.coe_add_monoid_hom_mk RingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ def copy (f : α →+* β) (f' : α → β) (h : f' = f) : α →+* β := { f.toMonoidWithZeroHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align ring_hom.copy RingHom.copy @[simp] theorem coe_copy (f : α →+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align ring_hom.coe_copy RingHom.coe_copy theorem copy_eq (f : α →+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align ring_hom.copy_eq RingHom.copy_eq end coe section variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} (f : α →+* β) {x y : α} theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x := FunLike.congr_fun h x #align ring_hom.congr_fun RingHom.congr_fun theorem congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y := FunLike.congr_arg f h #align ring_hom.congr_arg RingHom.congr_arg theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g := FunLike.coe_injective h #align ring_hom.coe_inj RingHom.coe_inj @[ext] theorem ext ⦃f g : α →+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align ring_hom.ext RingHom.ext theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align ring_hom.ext_iff RingHom.ext_iff @[simp] theorem mk_coe (f : α →+* β) (h₁ h₂ h₃ h₄) : RingHom.mk ⟨⟨f, h₁⟩, h₂⟩ h₃ h₄ = f := ext fun _ => rfl #align ring_hom.mk_coe RingHom.mk_coe theorem coe_addMonoidHom_injective : Injective (fun f : α →+* β => (f : α →+ β)) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align ring_hom.coe_add_monoid_hom_injective RingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_monoidHom_injective : Injective (fun f : α →+* β => (f : α →* β)) := fun _ _ h => ext <| MonoidHom.congr_fun h #align ring_hom.coe_monoid_hom_injective RingHom.coe_monoidHom_injective /-- Ring homomorphisms map zero to zero. -/ protected theorem map_zero (f : α →+* β) : f 0 = 0 := map_zero f #align ring_hom.map_zero RingHom.map_zero /-- Ring homomorphisms map one to one. -/ protected theorem map_one (f : α →+* β) : f 1 = 1 := map_one f #align ring_hom.map_one RingHom.map_one /-- Ring homomorphisms preserve addition. -/ protected theorem map_add (f : α →+* β) : ∀ a b, f (a + b) = f a + f b := map_add f #align ring_hom.map_add RingHom.map_add /-- Ring homomorphisms preserve multiplication. -/ protected theorem map_mul (f : α →+* β) : ∀ a b, f (a * b) = f a * f b := map_mul f #align ring_hom.map_mul RingHom.map_mul @[simp] theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 0 1) = ite p 0 1 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_zero_one RingHom.map_ite_zero_one @[simp] theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 1 0) = ite p 1 0 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_one_zero RingHom.map_ite_one_zero /-- `f : α →+* β` has a trivial codomain iff `f 1 = 0`. -/ theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 := by rw [map_one, eq_comm] #align ring_hom.codomain_trivial_iff_map_one_eq_zero RingHom.codomain_trivial_iff_map_one_eq_zero /-- `f : α →+* β` has a trivial codomain iff it has a trivial range. -/ theorem codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 := f.codomain_trivial_iff_map_one_eq_zero.trans ⟨fun h x => by rw [← mul_one x, map_mul, h, mul_zero], fun h => h 1⟩ #align ring_hom.codomain_trivial_iff_range_trivial RingHom.codomain_trivial_iff_range_trivial /-- `f : α →+* β` doesn't map `1` to `0` if `β` is nontrivial -/ theorem map_one_ne_zero [Nontrivial β] : f 1 ≠ 0 := mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one #align ring_hom.map_one_ne_zero RingHom.map_one_ne_zero /-- If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. -/ theorem domain_nontrivial [Nontrivial β] : Nontrivial α := ⟨⟨1, 0, mt (fun h => show f 1 = 0 by rw [h, map_zero]) f.map_one_ne_zero⟩⟩ #align ring_hom.domain_nontrivial RingHom.domain_nontrivial theorem codomain_trivial (f : α →+* β) [h : Subsingleton α] : Subsingleton β := (subsingleton_or_nontrivial β).resolve_right fun _ => not_nontrivial_iff_subsingleton.mpr h f.domain_nontrivial #align ring_hom.codomain_trivial RingHom.codomain_trivial end /-- Ring homomorphisms preserve additive inverse. -/ protected theorem map_neg [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x : α) : f (-x) = -f x := map_neg f x #align ring_hom.map_neg RingHom.map_neg /-- Ring homomorphisms preserve subtraction. -/ protected theorem map_sub [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x y : α) : f (x - y) = f x - f y := map_sub f x y #align ring_hom.map_sub RingHom.map_sub /-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/ def mk' [NonAssocSemiring α] [NonAssocRing β] (f : α →* β) (map_add : ∀ a b, f (a + b) = f a + f b) : α →+* β := { AddMonoidHom.mk' f map_add, f with } #align ring_hom.mk' RingHom.mk' variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} /-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;> intros <;> rfl #align ring_hom.id RingHom.id instance : Inhabited (α →+* α) := ⟨id α⟩ @[simp] theorem id_apply (x : α) : RingHom.id α x = x := rfl #align ring_hom.id_apply RingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (id α : α →+ α) = AddMonoidHom.id α := rfl #align ring_hom.coe_add_monoid_hom_id RingHom.coe_addMonoidHom_id @[simp] theorem coe_monoidHom_id : (id α : α →* α) = MonoidHom.id α := rfl #align ring_hom.coe_monoid_hom_id RingHom.coe_monoidHom_id variable {_ : NonAssocSemiring γ} /-- Composition of ring homomorphisms is a ring homomorphism. -/ def comp (g : β →+* γ) (f : α →+* β) : α →+* γ := { g.toNonUnitalRingHom.comp f.toNonUnitalRingHom with toFun := g ∘ f, map_one' := by simp } #align ring_hom.comp RingHom.comp /-- Composition of semiring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonAssocSemiring δ} (f : α →+* β) (g : β →+* γ) (h : γ →+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align ring_hom.comp_assoc RingHom.comp_assoc @[simp] theorem coe_comp (hnp : β →+* γ) (hmn : α →+* β) : (hnp.comp hmn : α → γ) = hnp ∘ hmn := rfl #align ring_hom.coe_comp RingHom.coe_comp theorem comp_apply (hnp : β →+* γ) (hmn : α →+* β) (x : α) : (hnp.comp hmn : α → γ) x = hnp (hmn x) := rfl #align ring_hom.comp_apply RingHom.comp_apply @[simp] theorem comp_id (f : α →+* β) : f.comp (id α) = f := ext fun _ => rfl #align ring_hom.comp_id RingHom.comp_id @[simp] theorem id_comp (f : α →+* β) : (id β).comp f = f := ext fun _ => rfl #align ring_hom.id_comp RingHom.id_comp instance : Monoid (α →+* α) where one := id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ theorem one_def : (1 : α →+* α) = id α := rfl #align ring_hom.one_def RingHom.one_def theorem mul_def (f g : α →+* α) : f * g = f.comp g := rfl #align ring_hom.mul_def RingHom.mul_def @[simp] theorem coe_one : ⇑(1 : α →+* α) = _root_.id := rfl #align ring_hom.coe_one RingHom.coe_one @[simp] theorem coe_mul (f g : α →+* α) : ⇑(f * g) = f ∘ g := rfl #align ring_hom.coe_mul RingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →+* γ} {f : α →+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => RingHom.ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align ring_hom.cancel_right RingHom.cancel_right @[simp] theorem cancel_left {g : β →+* γ} {f₁ f₂ : α →+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => RingHom.ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align ring_hom.cancel_left RingHom.cancel_left end RingHom namespace AddMonoidHom variable [CommRing α] [IsDomain α] [CommRing β] (f : β →+ α) -- Porting note: there's some disagreement over the naming scheme here. -- This could perhaps be `mkRingHom_of_mul_self_of_two_ne_zero`. -- See https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/naming.20conventions/near/315558410 /-- Make a ring homomorphism from an additive group homomorphism from a commutative ring to an integral domain that commutes with self multiplication, assumes that two is nonzero and `1` is sent to `1`. -/ def mkRingHomOfMulSelfOfTwoNeZero (h : ∀ x, f (x * x) = f x * f x) (h_two : (2 : α) ≠ 0) (h_one : f 1 = 1) : β →+* α := { f with map_one' := h_one, map_mul' := fun x y => by have hxy := h (x + y)

rw [mul_add, add_mul, add_mul, f.map_add, f.map_add, f.map_add, f.map_add, h x, h y, add_mul, mul_add, mul_add, ← sub_eq_zero, add_comm (f x * f x + f (y * x)), ← sub_sub, ← sub_sub, ← sub_sub, mul_comm y x, mul_comm (f y) (f x)] at hxy

/-- Make a ring homomorphism from an additive group homomorphism from a commutative ring to an integral domain that commutes with self multiplication, assumes that two is nonzero and `1` is sent to `1`. -/ def mkRingHomOfMulSelfOfTwoNeZero (h : ∀ x, f (x * x) = f x * f x) (h_two : (2 : α) ≠ 0) (h_one : f 1 = 1) : β →+* α := { f with map_one' := h_one, map_mul' := fun x y => by have hxy := h (x + y)

Mathlib.Algebra.Ring.Hom.Defs.733_0.KyHvVYrIs9pW9ZQ

/-- Make a ring homomorphism from an additive group homomorphism from a commutative ring to an integral domain that commutes with self multiplication, assumes that two is nonzero and `1` is sent to `1`. -/ def mkRingHomOfMulSelfOfTwoNeZero (h : ∀ x, f (x * x) = f x * f x) (h_two : (2 : α) ≠ 0) (h_one : f 1 = 1) : β →+* α

Mathlib_Algebra_Ring_Hom_Defs

F : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : CommRing α inst✝¹ : IsDomain α inst✝ : CommRing β f : β →+ α h : ∀ (x : β), f (x * x) = f x * f x h_two : 2 ≠ 0 h_one : f 1 = 1 x y : β hxy : f (x * y) + f y * f y + (f x * f x + f (x * y)) - f x * f x - f x * f y - f x * f y - f y * f y = 0 ⊢ OneHom.toFun { toFun := f.toFun, map_one' := h_one } (x * y) = OneHom.toFun { toFun := f.toFun, map_one' := h_one } x * OneHom.toFun { toFun := f.toFun, map_one' := h_one } y

/- Copyright (c) 2019 Amelia Livingston. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Amelia Livingston, Jireh Loreaux -/ import Mathlib.Algebra.Ring.Defs import Mathlib.Algebra.Ring.Basic import Mathlib.Data.Pi.Algebra #align_import algebra.hom.ring from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9" /-! # Homomorphisms of semirings and rings This file defines bundled homomorphisms of (non-unital) semirings and rings. As with monoid and groups, we use the same structure `RingHom a β`, a.k.a. `α →+* β`, for both types of homomorphisms. ## Main definitions * `NonUnitalRingHom`: Non-unital (semi)ring homomorphisms. Additive monoid homomorphism which preserve multiplication. * `RingHom`: (Semi)ring homomorphisms. Monoid homomorphisms which are also additive monoid homomorphism. ## Notations * `→ₙ+*`: Non-unital (semi)ring homs * `→+*`: (Semi)ring homs ## Implementation notes * There's a coercion from bundled homs to fun, and the canonical notation is to use the bundled hom as a function via this coercion. * There is no `SemiringHom` -- the idea is that `RingHom` is used. The constructor for a `RingHom` between semirings needs a proof of `map_zero`, `map_one` and `map_add` as well as `map_mul`; a separate constructor `RingHom.mk'` will construct ring homs between rings from monoid homs given only a proof that addition is preserved. ## Tags `RingHom`, `SemiringHom` -/ open Function variable {F α β γ : Type*} /-- Bundled non-unital semiring homomorphisms `α →ₙ+* β`; use this for bundled non-unital ring homomorphisms too. When possible, instead of parametrizing results over `(f : α →ₙ+* β)`, you should parametrize over `(F : Type*) [NonUnitalRingHomClass F α β] (f : F)`. When you extend this structure, make sure to extend `NonUnitalRingHomClass`. -/ structure NonUnitalRingHom (α β : Type*) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends α →ₙ* β, α →+ β #align non_unital_ring_hom NonUnitalRingHom /-- `α →ₙ+* β` denotes the type of non-unital ring homomorphisms from `α` to `β`. -/ infixr:25 " →ₙ+* " => NonUnitalRingHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as a semigroup homomorphism `α →ₙ* β`. The `simp`-normal form is `(f : α →ₙ* β)`. -/ add_decl_doc NonUnitalRingHom.toMulHom #align non_unital_ring_hom.to_mul_hom NonUnitalRingHom.toMulHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc NonUnitalRingHom.toAddMonoidHom #align non_unital_ring_hom.to_add_monoid_hom NonUnitalRingHom.toAddMonoidHom section NonUnitalRingHomClass /-- `NonUnitalRingHomClass F α β` states that `F` is a type of non-unital (semi)ring homomorphisms. You should extend this class when you extend `NonUnitalRingHom`. -/ class NonUnitalRingHomClass (F : Type*) (α β : outParam (Type*)) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends MulHomClass F α β, AddMonoidHomClass F α β #align non_unital_ring_hom_class NonUnitalRingHomClass variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] [NonUnitalRingHomClass F α β] /-- Turn an element of a type `F` satisfying `NonUnitalRingHomClass F α β` into an actual `NonUnitalRingHom`. This is declared as the default coercion from `F` to `α →ₙ+* β`. -/ @[coe] def NonUnitalRingHomClass.toNonUnitalRingHom (f : F) : α →ₙ+* β := { (f : α →ₙ* β), (f : α →+ β) with } /-- Any type satisfying `NonUnitalRingHomClass` can be cast into `NonUnitalRingHom` via `NonUnitalRingHomClass.toNonUnitalRingHom`. -/ instance : CoeTC F (α →ₙ+* β) := ⟨NonUnitalRingHomClass.toNonUnitalRingHom⟩ end NonUnitalRingHomClass namespace NonUnitalRingHom section coe variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] instance : NonUnitalRingHomClass (α →ₙ+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := NonUnitalRingHom.map_add' map_zero := NonUnitalRingHom.map_zero' map_mul f := f.map_mul' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -/ -- instance : CoeFun (α →ₙ+* β) fun _ => α → β := -- ⟨fun f => f.toFun⟩ -- Porting note: removed due to new `coe` in Lean4 #noalign non_unital_ring_hom.to_fun_eq_coe #noalign non_unital_ring_hom.coe_mk #noalign non_unital_ring_hom.coe_coe initialize_simps_projections NonUnitalRingHom (toFun → apply) @[simp] theorem coe_toMulHom (f : α →ₙ+* β) : ⇑f.toMulHom = f := rfl #align non_unital_ring_hom.coe_to_mul_hom NonUnitalRingHom.coe_toMulHom @[simp] theorem coe_mulHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →ₙ* β) = ⟨f, h₁⟩ := rfl #align non_unital_ring_hom.coe_mul_hom_mk NonUnitalRingHom.coe_mulHom_mk theorem coe_toAddMonoidHom (f : α →ₙ+* β) : ⇑f.toAddMonoidHom = f := rfl #align non_unital_ring_hom.coe_to_add_monoid_hom NonUnitalRingHom.coe_toAddMonoidHom @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →+ β) = ⟨⟨f, h₂⟩, h₃⟩ := rfl #align non_unital_ring_hom.coe_add_monoid_hom_mk NonUnitalRingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : α →ₙ+* β := { f.toMulHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align non_unital_ring_hom.copy NonUnitalRingHom.copy @[simp] theorem coe_copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align non_unital_ring_hom.coe_copy NonUnitalRingHom.coe_copy theorem copy_eq (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align non_unital_ring_hom.copy_eq NonUnitalRingHom.copy_eq end coe section variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] variable (f : α →ₙ+* β) {x y : α} @[ext] theorem ext ⦃f g : α →ₙ+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align non_unital_ring_hom.ext NonUnitalRingHom.ext theorem ext_iff {f g : α →ₙ+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align non_unital_ring_hom.ext_iff NonUnitalRingHom.ext_iff @[simp] theorem mk_coe (f : α →ₙ+* β) (h₁ h₂ h₃) : NonUnitalRingHom.mk (MulHom.mk f h₁) h₂ h₃ = f := ext fun _ => rfl #align non_unital_ring_hom.mk_coe NonUnitalRingHom.mk_coe theorem coe_addMonoidHom_injective : Injective fun f : α →ₙ+* β => (f : α →+ β) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align non_unital_ring_hom.coe_add_monoid_hom_injective NonUnitalRingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_mulHom_injective : Injective fun f : α →ₙ+* β => (f : α →ₙ* β) := fun _ _ h => ext <| MulHom.congr_fun h #align non_unital_ring_hom.coe_mul_hom_injective NonUnitalRingHom.coe_mulHom_injective end variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/ protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by refine' { toFun := id.. } <;> intros <;> rfl #align non_unital_ring_hom.id NonUnitalRingHom.id instance : Zero (α →ₙ+* β) := ⟨{ toFun := 0, map_mul' := fun _ _ => (mul_zero (0 : β)).symm, map_zero' := rfl, map_add' := fun _ _ => (add_zero (0 : β)).symm }⟩ instance : Inhabited (α →ₙ+* β) := ⟨0⟩ @[simp] theorem coe_zero : ⇑(0 : α →ₙ+* β) = 0 := rfl #align non_unital_ring_hom.coe_zero NonUnitalRingHom.coe_zero @[simp] theorem zero_apply (x : α) : (0 : α →ₙ+* β) x = 0 := rfl #align non_unital_ring_hom.zero_apply NonUnitalRingHom.zero_apply @[simp] theorem id_apply (x : α) : NonUnitalRingHom.id α x = x := rfl #align non_unital_ring_hom.id_apply NonUnitalRingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (NonUnitalRingHom.id α : α →+ α) = AddMonoidHom.id α := rfl #align non_unital_ring_hom.coe_add_monoid_hom_id NonUnitalRingHom.coe_addMonoidHom_id @[simp] theorem coe_mulHom_id : (NonUnitalRingHom.id α : α →ₙ* α) = MulHom.id α := rfl #align non_unital_ring_hom.coe_mul_hom_id NonUnitalRingHom.coe_mulHom_id variable [NonUnitalNonAssocSemiring γ] /-- Composition of non-unital ring homomorphisms is a non-unital ring homomorphism. -/ def comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : α →ₙ+* γ := { g.toMulHom.comp f.toMulHom, g.toAddMonoidHom.comp f.toAddMonoidHom with } #align non_unital_ring_hom.comp NonUnitalRingHom.comp /-- Composition of non-unital ring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonUnitalNonAssocSemiring δ} (f : α →ₙ+* β) (g : β →ₙ+* γ) (h : γ →ₙ+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align non_unital_ring_hom.comp_assoc NonUnitalRingHom.comp_assoc @[simp] theorem coe_comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : ⇑(g.comp f) = g ∘ f := rfl #align non_unital_ring_hom.coe_comp NonUnitalRingHom.coe_comp @[simp] theorem comp_apply (g : β →ₙ+* γ) (f : α →ₙ+* β) (x : α) : g.comp f x = g (f x) := rfl #align non_unital_ring_hom.comp_apply NonUnitalRingHom.comp_apply variable (g : β →ₙ+* γ) (f : α →ₙ+* β) @[simp] theorem coe_comp_addMonoidHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : AddMonoidHom.mk ⟨g ∘ f, (g.comp f).map_zero'⟩ (g.comp f).map_add' = (g : β →+ γ).comp f := rfl #align non_unital_ring_hom.coe_comp_add_monoid_hom NonUnitalRingHom.coe_comp_addMonoidHom @[simp] theorem coe_comp_mulHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : MulHom.mk (g ∘ f) (g.comp f).map_mul' = (g : β →ₙ* γ).comp f := rfl #align non_unital_ring_hom.coe_comp_mul_hom NonUnitalRingHom.coe_comp_mulHom @[simp] theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 := by ext simp #align non_unital_ring_hom.comp_zero NonUnitalRingHom.comp_zero @[simp] theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 := by ext rfl #align non_unital_ring_hom.zero_comp NonUnitalRingHom.zero_comp @[simp] theorem comp_id (f : α →ₙ+* β) : f.comp (NonUnitalRingHom.id α) = f := ext fun _ => rfl #align non_unital_ring_hom.comp_id NonUnitalRingHom.comp_id @[simp] theorem id_comp (f : α →ₙ+* β) : (NonUnitalRingHom.id β).comp f = f := ext fun _ => rfl #align non_unital_ring_hom.id_comp NonUnitalRingHom.id_comp instance : MonoidWithZero (α →ₙ+* α) where one := NonUnitalRingHom.id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ zero := 0 mul_zero := comp_zero zero_mul := zero_comp theorem one_def : (1 : α →ₙ+* α) = NonUnitalRingHom.id α := rfl #align non_unital_ring_hom.one_def NonUnitalRingHom.one_def @[simp] theorem coe_one : ⇑(1 : α →ₙ+* α) = id := rfl #align non_unital_ring_hom.coe_one NonUnitalRingHom.coe_one theorem mul_def (f g : α →ₙ+* α) : f * g = f.comp g := rfl #align non_unital_ring_hom.mul_def NonUnitalRingHom.mul_def @[simp] theorem coe_mul (f g : α →ₙ+* α) : ⇑(f * g) = f ∘ g := rfl #align non_unital_ring_hom.coe_mul NonUnitalRingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →ₙ+* γ} {f : α →ₙ+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_right NonUnitalRingHom.cancel_right @[simp] theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_left NonUnitalRingHom.cancel_left end NonUnitalRingHom /-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too. This extends from both `MonoidHom` and `MonoidWithZeroHom` in order to put the fields in a sensible order, even though `MonoidWithZeroHom` already extends `MonoidHom`. -/ structure RingHom (α : Type*) (β : Type*) [NonAssocSemiring α] [NonAssocSemiring β] extends α →* β, α →+ β, α →ₙ+* β, α →*₀ β #align ring_hom RingHom /-- `α →+* β` denotes the type of ring homomorphisms from `α` to `β`. -/ infixr:25 " →+* " => RingHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid with zero homomorphism `α →*₀ β`. The `simp`-normal form is `(f : α →*₀ β)`. -/ add_decl_doc RingHom.toMonoidWithZeroHom #align ring_hom.to_monoid_with_zero_hom RingHom.toMonoidWithZeroHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid homomorphism `α →* β`. The `simp`-normal form is `(f : α →* β)`. -/ add_decl_doc RingHom.toMonoidHom #align ring_hom.to_monoid_hom RingHom.toMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc RingHom.toAddMonoidHom #align ring_hom.to_add_monoid_hom RingHom.toAddMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a non-unital ring homomorphism `α →ₙ+* β`. The `simp`-normal form is `(f : α →ₙ+* β)`. -/ add_decl_doc RingHom.toNonUnitalRingHom #align ring_hom.to_non_unital_ring_hom RingHom.toNonUnitalRingHom section RingHomClass /-- `RingHomClass F α β` states that `F` is a type of (semi)ring homomorphisms. You should extend this class when you extend `RingHom`. This extends from both `MonoidHomClass` and `MonoidWithZeroHomClass` in order to put the fields in a sensible order, even though `MonoidWithZeroHomClass` already extends `MonoidHomClass`. -/ class RingHomClass (F : Type*) (α β : outParam (Type*)) [NonAssocSemiring α] [NonAssocSemiring β] extends MonoidHomClass F α β, AddMonoidHomClass F α β, MonoidWithZeroHomClass F α β #align ring_hom_class RingHomClass set_option linter.deprecated false in /-- Ring homomorphisms preserve `bit1`. -/ @[simp] lemma map_bit1 [NonAssocSemiring α] [NonAssocSemiring β] [RingHomClass F α β] (f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) := by simp [bit1] #align map_bit1 map_bit1 -- Porting note: marked `{}` rather than `[]` to prevent dangerous instances variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} [RingHomClass F α β] /-- Turn an element of a type `F` satisfying `RingHomClass F α β` into an actual `RingHom`. This is declared as the default coercion from `F` to `α →+* β`. -/ @[coe] def RingHomClass.toRingHom (f : F) : α →+* β := { (f : α →* β), (f : α →+ β) with } /-- Any type satisfying `RingHomClass` can be cast into `RingHom` via `RingHomClass.toRingHom`. -/ instance : CoeTC F (α →+* β) := ⟨RingHomClass.toRingHom⟩ instance (priority := 100) RingHomClass.toNonUnitalRingHomClass : NonUnitalRingHomClass F α β := { ‹RingHomClass F α β› with } #align ring_hom_class.to_non_unital_ring_hom_class RingHomClass.toNonUnitalRingHomClass end RingHomClass namespace RingHom section coe /-! Throughout this section, some `Semiring` arguments are specified with `{}` instead of `[]`. See note [implicit instance arguments]. -/ variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} instance instRingHomClass : RingHomClass (α →+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := RingHom.map_add' map_zero := RingHom.map_zero' map_mul f := f.map_mul' map_one f := f.map_one' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -- -/ -- instance : CoeFun (α →+* β) fun _ => α → β := -- ⟨RingHom.toFun⟩ initialize_simps_projections RingHom (toFun → apply) -- Porting note: is this lemma still needed in Lean4? -- Porting note: because `f.toFun` really means `f.toMonoidHom.toOneHom.toFun` and -- `toMonoidHom_eq_coe` wants to simplify `f.toMonoidHom` to `(↑f : M →* N)`, this can't -- be a simp lemma anymore -- @[simp] theorem toFun_eq_coe (f : α →+* β) : f.toFun = f := rfl #align ring_hom.to_fun_eq_coe RingHom.toFun_eq_coe @[simp] theorem coe_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α → β) = f := rfl #align ring_hom.coe_mk RingHom.coe_mk @[simp] theorem coe_coe {F : Type*} [RingHomClass F α β] (f : F) : ((f : α →+* β) : α → β) = f := rfl #align ring_hom.coe_coe RingHom.coe_coe attribute [coe] RingHom.toMonoidHom instance coeToMonoidHom : Coe (α →+* β) (α →* β) := ⟨RingHom.toMonoidHom⟩ #align ring_hom.has_coe_monoid_hom RingHom.coeToMonoidHom -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_monoid_hom @[simp] theorem toMonoidHom_eq_coe (f : α →+* β) : f.toMonoidHom = f := rfl #align ring_hom.to_monoid_hom_eq_coe RingHom.toMonoidHom_eq_coe -- Porting note: this can't be a simp lemma anymore -- @[simp] theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f := by rfl #align ring_hom.to_monoid_with_zero_hom_eq_coe RingHom.toMonoidWithZeroHom_eq_coe @[simp] theorem coe_monoidHom_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α →* β) = f := rfl #align ring_hom.coe_monoid_hom_mk RingHom.coe_monoidHom_mk -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_add_monoid_hom @[simp] theorem toAddMonoidHom_eq_coe (f : α →+* β) : f.toAddMonoidHom = f := rfl #align ring_hom.to_add_monoid_hom_eq_coe RingHom.toAddMonoidHom_eq_coe @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃ h₄) : ((⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨⟨f, h₃⟩, h₄⟩ := rfl #align ring_hom.coe_add_monoid_hom_mk RingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ def copy (f : α →+* β) (f' : α → β) (h : f' = f) : α →+* β := { f.toMonoidWithZeroHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align ring_hom.copy RingHom.copy @[simp] theorem coe_copy (f : α →+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align ring_hom.coe_copy RingHom.coe_copy theorem copy_eq (f : α →+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align ring_hom.copy_eq RingHom.copy_eq end coe section variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} (f : α →+* β) {x y : α} theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x := FunLike.congr_fun h x #align ring_hom.congr_fun RingHom.congr_fun theorem congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y := FunLike.congr_arg f h #align ring_hom.congr_arg RingHom.congr_arg theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g := FunLike.coe_injective h #align ring_hom.coe_inj RingHom.coe_inj @[ext] theorem ext ⦃f g : α →+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align ring_hom.ext RingHom.ext theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align ring_hom.ext_iff RingHom.ext_iff @[simp] theorem mk_coe (f : α →+* β) (h₁ h₂ h₃ h₄) : RingHom.mk ⟨⟨f, h₁⟩, h₂⟩ h₃ h₄ = f := ext fun _ => rfl #align ring_hom.mk_coe RingHom.mk_coe theorem coe_addMonoidHom_injective : Injective (fun f : α →+* β => (f : α →+ β)) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align ring_hom.coe_add_monoid_hom_injective RingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_monoidHom_injective : Injective (fun f : α →+* β => (f : α →* β)) := fun _ _ h => ext <| MonoidHom.congr_fun h #align ring_hom.coe_monoid_hom_injective RingHom.coe_monoidHom_injective /-- Ring homomorphisms map zero to zero. -/ protected theorem map_zero (f : α →+* β) : f 0 = 0 := map_zero f #align ring_hom.map_zero RingHom.map_zero /-- Ring homomorphisms map one to one. -/ protected theorem map_one (f : α →+* β) : f 1 = 1 := map_one f #align ring_hom.map_one RingHom.map_one /-- Ring homomorphisms preserve addition. -/ protected theorem map_add (f : α →+* β) : ∀ a b, f (a + b) = f a + f b := map_add f #align ring_hom.map_add RingHom.map_add /-- Ring homomorphisms preserve multiplication. -/ protected theorem map_mul (f : α →+* β) : ∀ a b, f (a * b) = f a * f b := map_mul f #align ring_hom.map_mul RingHom.map_mul @[simp] theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 0 1) = ite p 0 1 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_zero_one RingHom.map_ite_zero_one @[simp] theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 1 0) = ite p 1 0 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_one_zero RingHom.map_ite_one_zero /-- `f : α →+* β` has a trivial codomain iff `f 1 = 0`. -/ theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 := by rw [map_one, eq_comm] #align ring_hom.codomain_trivial_iff_map_one_eq_zero RingHom.codomain_trivial_iff_map_one_eq_zero /-- `f : α →+* β` has a trivial codomain iff it has a trivial range. -/ theorem codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 := f.codomain_trivial_iff_map_one_eq_zero.trans ⟨fun h x => by rw [← mul_one x, map_mul, h, mul_zero], fun h => h 1⟩ #align ring_hom.codomain_trivial_iff_range_trivial RingHom.codomain_trivial_iff_range_trivial /-- `f : α →+* β` doesn't map `1` to `0` if `β` is nontrivial -/ theorem map_one_ne_zero [Nontrivial β] : f 1 ≠ 0 := mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one #align ring_hom.map_one_ne_zero RingHom.map_one_ne_zero /-- If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. -/ theorem domain_nontrivial [Nontrivial β] : Nontrivial α := ⟨⟨1, 0, mt (fun h => show f 1 = 0 by rw [h, map_zero]) f.map_one_ne_zero⟩⟩ #align ring_hom.domain_nontrivial RingHom.domain_nontrivial theorem codomain_trivial (f : α →+* β) [h : Subsingleton α] : Subsingleton β := (subsingleton_or_nontrivial β).resolve_right fun _ => not_nontrivial_iff_subsingleton.mpr h f.domain_nontrivial #align ring_hom.codomain_trivial RingHom.codomain_trivial end /-- Ring homomorphisms preserve additive inverse. -/ protected theorem map_neg [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x : α) : f (-x) = -f x := map_neg f x #align ring_hom.map_neg RingHom.map_neg /-- Ring homomorphisms preserve subtraction. -/ protected theorem map_sub [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x y : α) : f (x - y) = f x - f y := map_sub f x y #align ring_hom.map_sub RingHom.map_sub /-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/ def mk' [NonAssocSemiring α] [NonAssocRing β] (f : α →* β) (map_add : ∀ a b, f (a + b) = f a + f b) : α →+* β := { AddMonoidHom.mk' f map_add, f with } #align ring_hom.mk' RingHom.mk' variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} /-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;> intros <;> rfl #align ring_hom.id RingHom.id instance : Inhabited (α →+* α) := ⟨id α⟩ @[simp] theorem id_apply (x : α) : RingHom.id α x = x := rfl #align ring_hom.id_apply RingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (id α : α →+ α) = AddMonoidHom.id α := rfl #align ring_hom.coe_add_monoid_hom_id RingHom.coe_addMonoidHom_id @[simp] theorem coe_monoidHom_id : (id α : α →* α) = MonoidHom.id α := rfl #align ring_hom.coe_monoid_hom_id RingHom.coe_monoidHom_id variable {_ : NonAssocSemiring γ} /-- Composition of ring homomorphisms is a ring homomorphism. -/ def comp (g : β →+* γ) (f : α →+* β) : α →+* γ := { g.toNonUnitalRingHom.comp f.toNonUnitalRingHom with toFun := g ∘ f, map_one' := by simp } #align ring_hom.comp RingHom.comp /-- Composition of semiring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonAssocSemiring δ} (f : α →+* β) (g : β →+* γ) (h : γ →+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align ring_hom.comp_assoc RingHom.comp_assoc @[simp] theorem coe_comp (hnp : β →+* γ) (hmn : α →+* β) : (hnp.comp hmn : α → γ) = hnp ∘ hmn := rfl #align ring_hom.coe_comp RingHom.coe_comp theorem comp_apply (hnp : β →+* γ) (hmn : α →+* β) (x : α) : (hnp.comp hmn : α → γ) x = hnp (hmn x) := rfl #align ring_hom.comp_apply RingHom.comp_apply @[simp] theorem comp_id (f : α →+* β) : f.comp (id α) = f := ext fun _ => rfl #align ring_hom.comp_id RingHom.comp_id @[simp] theorem id_comp (f : α →+* β) : (id β).comp f = f := ext fun _ => rfl #align ring_hom.id_comp RingHom.id_comp instance : Monoid (α →+* α) where one := id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ theorem one_def : (1 : α →+* α) = id α := rfl #align ring_hom.one_def RingHom.one_def theorem mul_def (f g : α →+* α) : f * g = f.comp g := rfl #align ring_hom.mul_def RingHom.mul_def @[simp] theorem coe_one : ⇑(1 : α →+* α) = _root_.id := rfl #align ring_hom.coe_one RingHom.coe_one @[simp] theorem coe_mul (f g : α →+* α) : ⇑(f * g) = f ∘ g := rfl #align ring_hom.coe_mul RingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →+* γ} {f : α →+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => RingHom.ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align ring_hom.cancel_right RingHom.cancel_right @[simp] theorem cancel_left {g : β →+* γ} {f₁ f₂ : α →+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => RingHom.ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align ring_hom.cancel_left RingHom.cancel_left end RingHom namespace AddMonoidHom variable [CommRing α] [IsDomain α] [CommRing β] (f : β →+ α) -- Porting note: there's some disagreement over the naming scheme here. -- This could perhaps be `mkRingHom_of_mul_self_of_two_ne_zero`. -- See https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/naming.20conventions/near/315558410 /-- Make a ring homomorphism from an additive group homomorphism from a commutative ring to an integral domain that commutes with self multiplication, assumes that two is nonzero and `1` is sent to `1`. -/ def mkRingHomOfMulSelfOfTwoNeZero (h : ∀ x, f (x * x) = f x * f x) (h_two : (2 : α) ≠ 0) (h_one : f 1 = 1) : β →+* α := { f with map_one' := h_one, map_mul' := fun x y => by have hxy := h (x + y) rw [mul_add, add_mul, add_mul, f.map_add, f.map_add, f.map_add, f.map_add, h x, h y, add_mul, mul_add, mul_add, ← sub_eq_zero, add_comm (f x * f x + f (y * x)), ← sub_sub, ← sub_sub, ← sub_sub, mul_comm y x, mul_comm (f y) (f x)] at hxy

simp only [add_assoc, add_sub_assoc, add_sub_cancel'_right] at hxy

/-- Make a ring homomorphism from an additive group homomorphism from a commutative ring to an integral domain that commutes with self multiplication, assumes that two is nonzero and `1` is sent to `1`. -/ def mkRingHomOfMulSelfOfTwoNeZero (h : ∀ x, f (x * x) = f x * f x) (h_two : (2 : α) ≠ 0) (h_one : f 1 = 1) : β →+* α := { f with map_one' := h_one, map_mul' := fun x y => by have hxy := h (x + y) rw [mul_add, add_mul, add_mul, f.map_add, f.map_add, f.map_add, f.map_add, h x, h y, add_mul, mul_add, mul_add, ← sub_eq_zero, add_comm (f x * f x + f (y * x)), ← sub_sub, ← sub_sub, ← sub_sub, mul_comm y x, mul_comm (f y) (f x)] at hxy

Mathlib.Algebra.Ring.Hom.Defs.733_0.KyHvVYrIs9pW9ZQ

/-- Make a ring homomorphism from an additive group homomorphism from a commutative ring to an integral domain that commutes with self multiplication, assumes that two is nonzero and `1` is sent to `1`. -/ def mkRingHomOfMulSelfOfTwoNeZero (h : ∀ x, f (x * x) = f x * f x) (h_two : (2 : α) ≠ 0) (h_one : f 1 = 1) : β →+* α

Mathlib_Algebra_Ring_Hom_Defs

F : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : CommRing α inst✝¹ : IsDomain α inst✝ : CommRing β f : β →+ α h : ∀ (x : β), f (x * x) = f x * f x h_two : 2 ≠ 0 h_one : f 1 = 1 x y : β hxy : f (x * y) + (f (x * y) - f x * f y - f x * f y) = 0 ⊢ OneHom.toFun { toFun := f.toFun, map_one' := h_one } (x * y) = OneHom.toFun { toFun := f.toFun, map_one' := h_one } x * OneHom.toFun { toFun := f.toFun, map_one' := h_one } y

/- Copyright (c) 2019 Amelia Livingston. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Amelia Livingston, Jireh Loreaux -/ import Mathlib.Algebra.Ring.Defs import Mathlib.Algebra.Ring.Basic import Mathlib.Data.Pi.Algebra #align_import algebra.hom.ring from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9" /-! # Homomorphisms of semirings and rings This file defines bundled homomorphisms of (non-unital) semirings and rings. As with monoid and groups, we use the same structure `RingHom a β`, a.k.a. `α →+* β`, for both types of homomorphisms. ## Main definitions * `NonUnitalRingHom`: Non-unital (semi)ring homomorphisms. Additive monoid homomorphism which preserve multiplication. * `RingHom`: (Semi)ring homomorphisms. Monoid homomorphisms which are also additive monoid homomorphism. ## Notations * `→ₙ+*`: Non-unital (semi)ring homs * `→+*`: (Semi)ring homs ## Implementation notes * There's a coercion from bundled homs to fun, and the canonical notation is to use the bundled hom as a function via this coercion. * There is no `SemiringHom` -- the idea is that `RingHom` is used. The constructor for a `RingHom` between semirings needs a proof of `map_zero`, `map_one` and `map_add` as well as `map_mul`; a separate constructor `RingHom.mk'` will construct ring homs between rings from monoid homs given only a proof that addition is preserved. ## Tags `RingHom`, `SemiringHom` -/ open Function variable {F α β γ : Type*} /-- Bundled non-unital semiring homomorphisms `α →ₙ+* β`; use this for bundled non-unital ring homomorphisms too. When possible, instead of parametrizing results over `(f : α →ₙ+* β)`, you should parametrize over `(F : Type*) [NonUnitalRingHomClass F α β] (f : F)`. When you extend this structure, make sure to extend `NonUnitalRingHomClass`. -/ structure NonUnitalRingHom (α β : Type*) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends α →ₙ* β, α →+ β #align non_unital_ring_hom NonUnitalRingHom /-- `α →ₙ+* β` denotes the type of non-unital ring homomorphisms from `α` to `β`. -/ infixr:25 " →ₙ+* " => NonUnitalRingHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as a semigroup homomorphism `α →ₙ* β`. The `simp`-normal form is `(f : α →ₙ* β)`. -/ add_decl_doc NonUnitalRingHom.toMulHom #align non_unital_ring_hom.to_mul_hom NonUnitalRingHom.toMulHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc NonUnitalRingHom.toAddMonoidHom #align non_unital_ring_hom.to_add_monoid_hom NonUnitalRingHom.toAddMonoidHom section NonUnitalRingHomClass /-- `NonUnitalRingHomClass F α β` states that `F` is a type of non-unital (semi)ring homomorphisms. You should extend this class when you extend `NonUnitalRingHom`. -/ class NonUnitalRingHomClass (F : Type*) (α β : outParam (Type*)) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends MulHomClass F α β, AddMonoidHomClass F α β #align non_unital_ring_hom_class NonUnitalRingHomClass variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] [NonUnitalRingHomClass F α β] /-- Turn an element of a type `F` satisfying `NonUnitalRingHomClass F α β` into an actual `NonUnitalRingHom`. This is declared as the default coercion from `F` to `α →ₙ+* β`. -/ @[coe] def NonUnitalRingHomClass.toNonUnitalRingHom (f : F) : α →ₙ+* β := { (f : α →ₙ* β), (f : α →+ β) with } /-- Any type satisfying `NonUnitalRingHomClass` can be cast into `NonUnitalRingHom` via `NonUnitalRingHomClass.toNonUnitalRingHom`. -/ instance : CoeTC F (α →ₙ+* β) := ⟨NonUnitalRingHomClass.toNonUnitalRingHom⟩ end NonUnitalRingHomClass namespace NonUnitalRingHom section coe variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] instance : NonUnitalRingHomClass (α →ₙ+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := NonUnitalRingHom.map_add' map_zero := NonUnitalRingHom.map_zero' map_mul f := f.map_mul' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -/ -- instance : CoeFun (α →ₙ+* β) fun _ => α → β := -- ⟨fun f => f.toFun⟩ -- Porting note: removed due to new `coe` in Lean4 #noalign non_unital_ring_hom.to_fun_eq_coe #noalign non_unital_ring_hom.coe_mk #noalign non_unital_ring_hom.coe_coe initialize_simps_projections NonUnitalRingHom (toFun → apply) @[simp] theorem coe_toMulHom (f : α →ₙ+* β) : ⇑f.toMulHom = f := rfl #align non_unital_ring_hom.coe_to_mul_hom NonUnitalRingHom.coe_toMulHom @[simp] theorem coe_mulHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →ₙ* β) = ⟨f, h₁⟩ := rfl #align non_unital_ring_hom.coe_mul_hom_mk NonUnitalRingHom.coe_mulHom_mk theorem coe_toAddMonoidHom (f : α →ₙ+* β) : ⇑f.toAddMonoidHom = f := rfl #align non_unital_ring_hom.coe_to_add_monoid_hom NonUnitalRingHom.coe_toAddMonoidHom @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →+ β) = ⟨⟨f, h₂⟩, h₃⟩ := rfl #align non_unital_ring_hom.coe_add_monoid_hom_mk NonUnitalRingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : α →ₙ+* β := { f.toMulHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align non_unital_ring_hom.copy NonUnitalRingHom.copy @[simp] theorem coe_copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align non_unital_ring_hom.coe_copy NonUnitalRingHom.coe_copy theorem copy_eq (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align non_unital_ring_hom.copy_eq NonUnitalRingHom.copy_eq end coe section variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] variable (f : α →ₙ+* β) {x y : α} @[ext] theorem ext ⦃f g : α →ₙ+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align non_unital_ring_hom.ext NonUnitalRingHom.ext theorem ext_iff {f g : α →ₙ+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align non_unital_ring_hom.ext_iff NonUnitalRingHom.ext_iff @[simp] theorem mk_coe (f : α →ₙ+* β) (h₁ h₂ h₃) : NonUnitalRingHom.mk (MulHom.mk f h₁) h₂ h₃ = f := ext fun _ => rfl #align non_unital_ring_hom.mk_coe NonUnitalRingHom.mk_coe theorem coe_addMonoidHom_injective : Injective fun f : α →ₙ+* β => (f : α →+ β) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align non_unital_ring_hom.coe_add_monoid_hom_injective NonUnitalRingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_mulHom_injective : Injective fun f : α →ₙ+* β => (f : α →ₙ* β) := fun _ _ h => ext <| MulHom.congr_fun h #align non_unital_ring_hom.coe_mul_hom_injective NonUnitalRingHom.coe_mulHom_injective end variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/ protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by refine' { toFun := id.. } <;> intros <;> rfl #align non_unital_ring_hom.id NonUnitalRingHom.id instance : Zero (α →ₙ+* β) := ⟨{ toFun := 0, map_mul' := fun _ _ => (mul_zero (0 : β)).symm, map_zero' := rfl, map_add' := fun _ _ => (add_zero (0 : β)).symm }⟩ instance : Inhabited (α →ₙ+* β) := ⟨0⟩ @[simp] theorem coe_zero : ⇑(0 : α →ₙ+* β) = 0 := rfl #align non_unital_ring_hom.coe_zero NonUnitalRingHom.coe_zero @[simp] theorem zero_apply (x : α) : (0 : α →ₙ+* β) x = 0 := rfl #align non_unital_ring_hom.zero_apply NonUnitalRingHom.zero_apply @[simp] theorem id_apply (x : α) : NonUnitalRingHom.id α x = x := rfl #align non_unital_ring_hom.id_apply NonUnitalRingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (NonUnitalRingHom.id α : α →+ α) = AddMonoidHom.id α := rfl #align non_unital_ring_hom.coe_add_monoid_hom_id NonUnitalRingHom.coe_addMonoidHom_id @[simp] theorem coe_mulHom_id : (NonUnitalRingHom.id α : α →ₙ* α) = MulHom.id α := rfl #align non_unital_ring_hom.coe_mul_hom_id NonUnitalRingHom.coe_mulHom_id variable [NonUnitalNonAssocSemiring γ] /-- Composition of non-unital ring homomorphisms is a non-unital ring homomorphism. -/ def comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : α →ₙ+* γ := { g.toMulHom.comp f.toMulHom, g.toAddMonoidHom.comp f.toAddMonoidHom with } #align non_unital_ring_hom.comp NonUnitalRingHom.comp /-- Composition of non-unital ring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonUnitalNonAssocSemiring δ} (f : α →ₙ+* β) (g : β →ₙ+* γ) (h : γ →ₙ+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align non_unital_ring_hom.comp_assoc NonUnitalRingHom.comp_assoc @[simp] theorem coe_comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : ⇑(g.comp f) = g ∘ f := rfl #align non_unital_ring_hom.coe_comp NonUnitalRingHom.coe_comp @[simp] theorem comp_apply (g : β →ₙ+* γ) (f : α →ₙ+* β) (x : α) : g.comp f x = g (f x) := rfl #align non_unital_ring_hom.comp_apply NonUnitalRingHom.comp_apply variable (g : β →ₙ+* γ) (f : α →ₙ+* β) @[simp] theorem coe_comp_addMonoidHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : AddMonoidHom.mk ⟨g ∘ f, (g.comp f).map_zero'⟩ (g.comp f).map_add' = (g : β →+ γ).comp f := rfl #align non_unital_ring_hom.coe_comp_add_monoid_hom NonUnitalRingHom.coe_comp_addMonoidHom @[simp] theorem coe_comp_mulHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : MulHom.mk (g ∘ f) (g.comp f).map_mul' = (g : β →ₙ* γ).comp f := rfl #align non_unital_ring_hom.coe_comp_mul_hom NonUnitalRingHom.coe_comp_mulHom @[simp] theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 := by ext simp #align non_unital_ring_hom.comp_zero NonUnitalRingHom.comp_zero @[simp] theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 := by ext rfl #align non_unital_ring_hom.zero_comp NonUnitalRingHom.zero_comp @[simp] theorem comp_id (f : α →ₙ+* β) : f.comp (NonUnitalRingHom.id α) = f := ext fun _ => rfl #align non_unital_ring_hom.comp_id NonUnitalRingHom.comp_id @[simp] theorem id_comp (f : α →ₙ+* β) : (NonUnitalRingHom.id β).comp f = f := ext fun _ => rfl #align non_unital_ring_hom.id_comp NonUnitalRingHom.id_comp instance : MonoidWithZero (α →ₙ+* α) where one := NonUnitalRingHom.id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ zero := 0 mul_zero := comp_zero zero_mul := zero_comp theorem one_def : (1 : α →ₙ+* α) = NonUnitalRingHom.id α := rfl #align non_unital_ring_hom.one_def NonUnitalRingHom.one_def @[simp] theorem coe_one : ⇑(1 : α →ₙ+* α) = id := rfl #align non_unital_ring_hom.coe_one NonUnitalRingHom.coe_one theorem mul_def (f g : α →ₙ+* α) : f * g = f.comp g := rfl #align non_unital_ring_hom.mul_def NonUnitalRingHom.mul_def @[simp] theorem coe_mul (f g : α →ₙ+* α) : ⇑(f * g) = f ∘ g := rfl #align non_unital_ring_hom.coe_mul NonUnitalRingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →ₙ+* γ} {f : α →ₙ+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_right NonUnitalRingHom.cancel_right @[simp] theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_left NonUnitalRingHom.cancel_left end NonUnitalRingHom /-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too. This extends from both `MonoidHom` and `MonoidWithZeroHom` in order to put the fields in a sensible order, even though `MonoidWithZeroHom` already extends `MonoidHom`. -/ structure RingHom (α : Type*) (β : Type*) [NonAssocSemiring α] [NonAssocSemiring β] extends α →* β, α →+ β, α →ₙ+* β, α →*₀ β #align ring_hom RingHom /-- `α →+* β` denotes the type of ring homomorphisms from `α` to `β`. -/ infixr:25 " →+* " => RingHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid with zero homomorphism `α →*₀ β`. The `simp`-normal form is `(f : α →*₀ β)`. -/ add_decl_doc RingHom.toMonoidWithZeroHom #align ring_hom.to_monoid_with_zero_hom RingHom.toMonoidWithZeroHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid homomorphism `α →* β`. The `simp`-normal form is `(f : α →* β)`. -/ add_decl_doc RingHom.toMonoidHom #align ring_hom.to_monoid_hom RingHom.toMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc RingHom.toAddMonoidHom #align ring_hom.to_add_monoid_hom RingHom.toAddMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a non-unital ring homomorphism `α →ₙ+* β`. The `simp`-normal form is `(f : α →ₙ+* β)`. -/ add_decl_doc RingHom.toNonUnitalRingHom #align ring_hom.to_non_unital_ring_hom RingHom.toNonUnitalRingHom section RingHomClass /-- `RingHomClass F α β` states that `F` is a type of (semi)ring homomorphisms. You should extend this class when you extend `RingHom`. This extends from both `MonoidHomClass` and `MonoidWithZeroHomClass` in order to put the fields in a sensible order, even though `MonoidWithZeroHomClass` already extends `MonoidHomClass`. -/ class RingHomClass (F : Type*) (α β : outParam (Type*)) [NonAssocSemiring α] [NonAssocSemiring β] extends MonoidHomClass F α β, AddMonoidHomClass F α β, MonoidWithZeroHomClass F α β #align ring_hom_class RingHomClass set_option linter.deprecated false in /-- Ring homomorphisms preserve `bit1`. -/ @[simp] lemma map_bit1 [NonAssocSemiring α] [NonAssocSemiring β] [RingHomClass F α β] (f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) := by simp [bit1] #align map_bit1 map_bit1 -- Porting note: marked `{}` rather than `[]` to prevent dangerous instances variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} [RingHomClass F α β] /-- Turn an element of a type `F` satisfying `RingHomClass F α β` into an actual `RingHom`. This is declared as the default coercion from `F` to `α →+* β`. -/ @[coe] def RingHomClass.toRingHom (f : F) : α →+* β := { (f : α →* β), (f : α →+ β) with } /-- Any type satisfying `RingHomClass` can be cast into `RingHom` via `RingHomClass.toRingHom`. -/ instance : CoeTC F (α →+* β) := ⟨RingHomClass.toRingHom⟩ instance (priority := 100) RingHomClass.toNonUnitalRingHomClass : NonUnitalRingHomClass F α β := { ‹RingHomClass F α β› with } #align ring_hom_class.to_non_unital_ring_hom_class RingHomClass.toNonUnitalRingHomClass end RingHomClass namespace RingHom section coe /-! Throughout this section, some `Semiring` arguments are specified with `{}` instead of `[]`. See note [implicit instance arguments]. -/ variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} instance instRingHomClass : RingHomClass (α →+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := RingHom.map_add' map_zero := RingHom.map_zero' map_mul f := f.map_mul' map_one f := f.map_one' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -- -/ -- instance : CoeFun (α →+* β) fun _ => α → β := -- ⟨RingHom.toFun⟩ initialize_simps_projections RingHom (toFun → apply) -- Porting note: is this lemma still needed in Lean4? -- Porting note: because `f.toFun` really means `f.toMonoidHom.toOneHom.toFun` and -- `toMonoidHom_eq_coe` wants to simplify `f.toMonoidHom` to `(↑f : M →* N)`, this can't -- be a simp lemma anymore -- @[simp] theorem toFun_eq_coe (f : α →+* β) : f.toFun = f := rfl #align ring_hom.to_fun_eq_coe RingHom.toFun_eq_coe @[simp] theorem coe_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α → β) = f := rfl #align ring_hom.coe_mk RingHom.coe_mk @[simp] theorem coe_coe {F : Type*} [RingHomClass F α β] (f : F) : ((f : α →+* β) : α → β) = f := rfl #align ring_hom.coe_coe RingHom.coe_coe attribute [coe] RingHom.toMonoidHom instance coeToMonoidHom : Coe (α →+* β) (α →* β) := ⟨RingHom.toMonoidHom⟩ #align ring_hom.has_coe_monoid_hom RingHom.coeToMonoidHom -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_monoid_hom @[simp] theorem toMonoidHom_eq_coe (f : α →+* β) : f.toMonoidHom = f := rfl #align ring_hom.to_monoid_hom_eq_coe RingHom.toMonoidHom_eq_coe -- Porting note: this can't be a simp lemma anymore -- @[simp] theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f := by rfl #align ring_hom.to_monoid_with_zero_hom_eq_coe RingHom.toMonoidWithZeroHom_eq_coe @[simp] theorem coe_monoidHom_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α →* β) = f := rfl #align ring_hom.coe_monoid_hom_mk RingHom.coe_monoidHom_mk -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_add_monoid_hom @[simp] theorem toAddMonoidHom_eq_coe (f : α →+* β) : f.toAddMonoidHom = f := rfl #align ring_hom.to_add_monoid_hom_eq_coe RingHom.toAddMonoidHom_eq_coe @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃ h₄) : ((⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨⟨f, h₃⟩, h₄⟩ := rfl #align ring_hom.coe_add_monoid_hom_mk RingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ def copy (f : α →+* β) (f' : α → β) (h : f' = f) : α →+* β := { f.toMonoidWithZeroHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align ring_hom.copy RingHom.copy @[simp] theorem coe_copy (f : α →+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align ring_hom.coe_copy RingHom.coe_copy theorem copy_eq (f : α →+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align ring_hom.copy_eq RingHom.copy_eq end coe section variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} (f : α →+* β) {x y : α} theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x := FunLike.congr_fun h x #align ring_hom.congr_fun RingHom.congr_fun theorem congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y := FunLike.congr_arg f h #align ring_hom.congr_arg RingHom.congr_arg theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g := FunLike.coe_injective h #align ring_hom.coe_inj RingHom.coe_inj @[ext] theorem ext ⦃f g : α →+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align ring_hom.ext RingHom.ext theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align ring_hom.ext_iff RingHom.ext_iff @[simp] theorem mk_coe (f : α →+* β) (h₁ h₂ h₃ h₄) : RingHom.mk ⟨⟨f, h₁⟩, h₂⟩ h₃ h₄ = f := ext fun _ => rfl #align ring_hom.mk_coe RingHom.mk_coe theorem coe_addMonoidHom_injective : Injective (fun f : α →+* β => (f : α →+ β)) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align ring_hom.coe_add_monoid_hom_injective RingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_monoidHom_injective : Injective (fun f : α →+* β => (f : α →* β)) := fun _ _ h => ext <| MonoidHom.congr_fun h #align ring_hom.coe_monoid_hom_injective RingHom.coe_monoidHom_injective /-- Ring homomorphisms map zero to zero. -/ protected theorem map_zero (f : α →+* β) : f 0 = 0 := map_zero f #align ring_hom.map_zero RingHom.map_zero /-- Ring homomorphisms map one to one. -/ protected theorem map_one (f : α →+* β) : f 1 = 1 := map_one f #align ring_hom.map_one RingHom.map_one /-- Ring homomorphisms preserve addition. -/ protected theorem map_add (f : α →+* β) : ∀ a b, f (a + b) = f a + f b := map_add f #align ring_hom.map_add RingHom.map_add /-- Ring homomorphisms preserve multiplication. -/ protected theorem map_mul (f : α →+* β) : ∀ a b, f (a * b) = f a * f b := map_mul f #align ring_hom.map_mul RingHom.map_mul @[simp] theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 0 1) = ite p 0 1 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_zero_one RingHom.map_ite_zero_one @[simp] theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 1 0) = ite p 1 0 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_one_zero RingHom.map_ite_one_zero /-- `f : α →+* β` has a trivial codomain iff `f 1 = 0`. -/ theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 := by rw [map_one, eq_comm] #align ring_hom.codomain_trivial_iff_map_one_eq_zero RingHom.codomain_trivial_iff_map_one_eq_zero /-- `f : α →+* β` has a trivial codomain iff it has a trivial range. -/ theorem codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 := f.codomain_trivial_iff_map_one_eq_zero.trans ⟨fun h x => by rw [← mul_one x, map_mul, h, mul_zero], fun h => h 1⟩ #align ring_hom.codomain_trivial_iff_range_trivial RingHom.codomain_trivial_iff_range_trivial /-- `f : α →+* β` doesn't map `1` to `0` if `β` is nontrivial -/ theorem map_one_ne_zero [Nontrivial β] : f 1 ≠ 0 := mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one #align ring_hom.map_one_ne_zero RingHom.map_one_ne_zero /-- If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. -/ theorem domain_nontrivial [Nontrivial β] : Nontrivial α := ⟨⟨1, 0, mt (fun h => show f 1 = 0 by rw [h, map_zero]) f.map_one_ne_zero⟩⟩ #align ring_hom.domain_nontrivial RingHom.domain_nontrivial theorem codomain_trivial (f : α →+* β) [h : Subsingleton α] : Subsingleton β := (subsingleton_or_nontrivial β).resolve_right fun _ => not_nontrivial_iff_subsingleton.mpr h f.domain_nontrivial #align ring_hom.codomain_trivial RingHom.codomain_trivial end /-- Ring homomorphisms preserve additive inverse. -/ protected theorem map_neg [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x : α) : f (-x) = -f x := map_neg f x #align ring_hom.map_neg RingHom.map_neg /-- Ring homomorphisms preserve subtraction. -/ protected theorem map_sub [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x y : α) : f (x - y) = f x - f y := map_sub f x y #align ring_hom.map_sub RingHom.map_sub /-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/ def mk' [NonAssocSemiring α] [NonAssocRing β] (f : α →* β) (map_add : ∀ a b, f (a + b) = f a + f b) : α →+* β := { AddMonoidHom.mk' f map_add, f with } #align ring_hom.mk' RingHom.mk' variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} /-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;> intros <;> rfl #align ring_hom.id RingHom.id instance : Inhabited (α →+* α) := ⟨id α⟩ @[simp] theorem id_apply (x : α) : RingHom.id α x = x := rfl #align ring_hom.id_apply RingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (id α : α →+ α) = AddMonoidHom.id α := rfl #align ring_hom.coe_add_monoid_hom_id RingHom.coe_addMonoidHom_id @[simp] theorem coe_monoidHom_id : (id α : α →* α) = MonoidHom.id α := rfl #align ring_hom.coe_monoid_hom_id RingHom.coe_monoidHom_id variable {_ : NonAssocSemiring γ} /-- Composition of ring homomorphisms is a ring homomorphism. -/ def comp (g : β →+* γ) (f : α →+* β) : α →+* γ := { g.toNonUnitalRingHom.comp f.toNonUnitalRingHom with toFun := g ∘ f, map_one' := by simp } #align ring_hom.comp RingHom.comp /-- Composition of semiring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonAssocSemiring δ} (f : α →+* β) (g : β →+* γ) (h : γ →+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align ring_hom.comp_assoc RingHom.comp_assoc @[simp] theorem coe_comp (hnp : β →+* γ) (hmn : α →+* β) : (hnp.comp hmn : α → γ) = hnp ∘ hmn := rfl #align ring_hom.coe_comp RingHom.coe_comp theorem comp_apply (hnp : β →+* γ) (hmn : α →+* β) (x : α) : (hnp.comp hmn : α → γ) x = hnp (hmn x) := rfl #align ring_hom.comp_apply RingHom.comp_apply @[simp] theorem comp_id (f : α →+* β) : f.comp (id α) = f := ext fun _ => rfl #align ring_hom.comp_id RingHom.comp_id @[simp] theorem id_comp (f : α →+* β) : (id β).comp f = f := ext fun _ => rfl #align ring_hom.id_comp RingHom.id_comp instance : Monoid (α →+* α) where one := id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ theorem one_def : (1 : α →+* α) = id α := rfl #align ring_hom.one_def RingHom.one_def theorem mul_def (f g : α →+* α) : f * g = f.comp g := rfl #align ring_hom.mul_def RingHom.mul_def @[simp] theorem coe_one : ⇑(1 : α →+* α) = _root_.id := rfl #align ring_hom.coe_one RingHom.coe_one @[simp] theorem coe_mul (f g : α →+* α) : ⇑(f * g) = f ∘ g := rfl #align ring_hom.coe_mul RingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →+* γ} {f : α →+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => RingHom.ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align ring_hom.cancel_right RingHom.cancel_right @[simp] theorem cancel_left {g : β →+* γ} {f₁ f₂ : α →+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => RingHom.ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align ring_hom.cancel_left RingHom.cancel_left end RingHom namespace AddMonoidHom variable [CommRing α] [IsDomain α] [CommRing β] (f : β →+ α) -- Porting note: there's some disagreement over the naming scheme here. -- This could perhaps be `mkRingHom_of_mul_self_of_two_ne_zero`. -- See https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/naming.20conventions/near/315558410 /-- Make a ring homomorphism from an additive group homomorphism from a commutative ring to an integral domain that commutes with self multiplication, assumes that two is nonzero and `1` is sent to `1`. -/ def mkRingHomOfMulSelfOfTwoNeZero (h : ∀ x, f (x * x) = f x * f x) (h_two : (2 : α) ≠ 0) (h_one : f 1 = 1) : β →+* α := { f with map_one' := h_one, map_mul' := fun x y => by have hxy := h (x + y) rw [mul_add, add_mul, add_mul, f.map_add, f.map_add, f.map_add, f.map_add, h x, h y, add_mul, mul_add, mul_add, ← sub_eq_zero, add_comm (f x * f x + f (y * x)), ← sub_sub, ← sub_sub, ← sub_sub, mul_comm y x, mul_comm (f y) (f x)] at hxy simp only [add_assoc, add_sub_assoc, add_sub_cancel'_right] at hxy

rw [sub_sub, ← two_mul, ← add_sub_assoc, ← two_mul, ← mul_sub, mul_eq_zero (M₀ := α), sub_eq_zero, or_iff_not_imp_left] at hxy

/-- Make a ring homomorphism from an additive group homomorphism from a commutative ring to an integral domain that commutes with self multiplication, assumes that two is nonzero and `1` is sent to `1`. -/ def mkRingHomOfMulSelfOfTwoNeZero (h : ∀ x, f (x * x) = f x * f x) (h_two : (2 : α) ≠ 0) (h_one : f 1 = 1) : β →+* α := { f with map_one' := h_one, map_mul' := fun x y => by have hxy := h (x + y) rw [mul_add, add_mul, add_mul, f.map_add, f.map_add, f.map_add, f.map_add, h x, h y, add_mul, mul_add, mul_add, ← sub_eq_zero, add_comm (f x * f x + f (y * x)), ← sub_sub, ← sub_sub, ← sub_sub, mul_comm y x, mul_comm (f y) (f x)] at hxy simp only [add_assoc, add_sub_assoc, add_sub_cancel'_right] at hxy

Mathlib.Algebra.Ring.Hom.Defs.733_0.KyHvVYrIs9pW9ZQ

/-- Make a ring homomorphism from an additive group homomorphism from a commutative ring to an integral domain that commutes with self multiplication, assumes that two is nonzero and `1` is sent to `1`. -/ def mkRingHomOfMulSelfOfTwoNeZero (h : ∀ x, f (x * x) = f x * f x) (h_two : (2 : α) ≠ 0) (h_one : f 1 = 1) : β →+* α

Mathlib_Algebra_Ring_Hom_Defs

F : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : CommRing α inst✝¹ : IsDomain α inst✝ : CommRing β f : β →+ α h : ∀ (x : β), f (x * x) = f x * f x h_two : 2 ≠ 0 h_one : f 1 = 1 x y : β hxy : ¬2 = 0 → f (x * y) = f x * f y ⊢ OneHom.toFun { toFun := f.toFun, map_one' := h_one } (x * y) = OneHom.toFun { toFun := f.toFun, map_one' := h_one } x * OneHom.toFun { toFun := f.toFun, map_one' := h_one } y

/- Copyright (c) 2019 Amelia Livingston. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Amelia Livingston, Jireh Loreaux -/ import Mathlib.Algebra.Ring.Defs import Mathlib.Algebra.Ring.Basic import Mathlib.Data.Pi.Algebra #align_import algebra.hom.ring from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9" /-! # Homomorphisms of semirings and rings This file defines bundled homomorphisms of (non-unital) semirings and rings. As with monoid and groups, we use the same structure `RingHom a β`, a.k.a. `α →+* β`, for both types of homomorphisms. ## Main definitions * `NonUnitalRingHom`: Non-unital (semi)ring homomorphisms. Additive monoid homomorphism which preserve multiplication. * `RingHom`: (Semi)ring homomorphisms. Monoid homomorphisms which are also additive monoid homomorphism. ## Notations * `→ₙ+*`: Non-unital (semi)ring homs * `→+*`: (Semi)ring homs ## Implementation notes * There's a coercion from bundled homs to fun, and the canonical notation is to use the bundled hom as a function via this coercion. * There is no `SemiringHom` -- the idea is that `RingHom` is used. The constructor for a `RingHom` between semirings needs a proof of `map_zero`, `map_one` and `map_add` as well as `map_mul`; a separate constructor `RingHom.mk'` will construct ring homs between rings from monoid homs given only a proof that addition is preserved. ## Tags `RingHom`, `SemiringHom` -/ open Function variable {F α β γ : Type*} /-- Bundled non-unital semiring homomorphisms `α →ₙ+* β`; use this for bundled non-unital ring homomorphisms too. When possible, instead of parametrizing results over `(f : α →ₙ+* β)`, you should parametrize over `(F : Type*) [NonUnitalRingHomClass F α β] (f : F)`. When you extend this structure, make sure to extend `NonUnitalRingHomClass`. -/ structure NonUnitalRingHom (α β : Type*) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends α →ₙ* β, α →+ β #align non_unital_ring_hom NonUnitalRingHom /-- `α →ₙ+* β` denotes the type of non-unital ring homomorphisms from `α` to `β`. -/ infixr:25 " →ₙ+* " => NonUnitalRingHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as a semigroup homomorphism `α →ₙ* β`. The `simp`-normal form is `(f : α →ₙ* β)`. -/ add_decl_doc NonUnitalRingHom.toMulHom #align non_unital_ring_hom.to_mul_hom NonUnitalRingHom.toMulHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc NonUnitalRingHom.toAddMonoidHom #align non_unital_ring_hom.to_add_monoid_hom NonUnitalRingHom.toAddMonoidHom section NonUnitalRingHomClass /-- `NonUnitalRingHomClass F α β` states that `F` is a type of non-unital (semi)ring homomorphisms. You should extend this class when you extend `NonUnitalRingHom`. -/ class NonUnitalRingHomClass (F : Type*) (α β : outParam (Type*)) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends MulHomClass F α β, AddMonoidHomClass F α β #align non_unital_ring_hom_class NonUnitalRingHomClass variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] [NonUnitalRingHomClass F α β] /-- Turn an element of a type `F` satisfying `NonUnitalRingHomClass F α β` into an actual `NonUnitalRingHom`. This is declared as the default coercion from `F` to `α →ₙ+* β`. -/ @[coe] def NonUnitalRingHomClass.toNonUnitalRingHom (f : F) : α →ₙ+* β := { (f : α →ₙ* β), (f : α →+ β) with } /-- Any type satisfying `NonUnitalRingHomClass` can be cast into `NonUnitalRingHom` via `NonUnitalRingHomClass.toNonUnitalRingHom`. -/ instance : CoeTC F (α →ₙ+* β) := ⟨NonUnitalRingHomClass.toNonUnitalRingHom⟩ end NonUnitalRingHomClass namespace NonUnitalRingHom section coe variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] instance : NonUnitalRingHomClass (α →ₙ+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := NonUnitalRingHom.map_add' map_zero := NonUnitalRingHom.map_zero' map_mul f := f.map_mul' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -/ -- instance : CoeFun (α →ₙ+* β) fun _ => α → β := -- ⟨fun f => f.toFun⟩ -- Porting note: removed due to new `coe` in Lean4 #noalign non_unital_ring_hom.to_fun_eq_coe #noalign non_unital_ring_hom.coe_mk #noalign non_unital_ring_hom.coe_coe initialize_simps_projections NonUnitalRingHom (toFun → apply) @[simp] theorem coe_toMulHom (f : α →ₙ+* β) : ⇑f.toMulHom = f := rfl #align non_unital_ring_hom.coe_to_mul_hom NonUnitalRingHom.coe_toMulHom @[simp] theorem coe_mulHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →ₙ* β) = ⟨f, h₁⟩ := rfl #align non_unital_ring_hom.coe_mul_hom_mk NonUnitalRingHom.coe_mulHom_mk theorem coe_toAddMonoidHom (f : α →ₙ+* β) : ⇑f.toAddMonoidHom = f := rfl #align non_unital_ring_hom.coe_to_add_monoid_hom NonUnitalRingHom.coe_toAddMonoidHom @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →+ β) = ⟨⟨f, h₂⟩, h₃⟩ := rfl #align non_unital_ring_hom.coe_add_monoid_hom_mk NonUnitalRingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : α →ₙ+* β := { f.toMulHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align non_unital_ring_hom.copy NonUnitalRingHom.copy @[simp] theorem coe_copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align non_unital_ring_hom.coe_copy NonUnitalRingHom.coe_copy theorem copy_eq (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align non_unital_ring_hom.copy_eq NonUnitalRingHom.copy_eq end coe section variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] variable (f : α →ₙ+* β) {x y : α} @[ext] theorem ext ⦃f g : α →ₙ+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align non_unital_ring_hom.ext NonUnitalRingHom.ext theorem ext_iff {f g : α →ₙ+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align non_unital_ring_hom.ext_iff NonUnitalRingHom.ext_iff @[simp] theorem mk_coe (f : α →ₙ+* β) (h₁ h₂ h₃) : NonUnitalRingHom.mk (MulHom.mk f h₁) h₂ h₃ = f := ext fun _ => rfl #align non_unital_ring_hom.mk_coe NonUnitalRingHom.mk_coe theorem coe_addMonoidHom_injective : Injective fun f : α →ₙ+* β => (f : α →+ β) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align non_unital_ring_hom.coe_add_monoid_hom_injective NonUnitalRingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_mulHom_injective : Injective fun f : α →ₙ+* β => (f : α →ₙ* β) := fun _ _ h => ext <| MulHom.congr_fun h #align non_unital_ring_hom.coe_mul_hom_injective NonUnitalRingHom.coe_mulHom_injective end variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/ protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by refine' { toFun := id.. } <;> intros <;> rfl #align non_unital_ring_hom.id NonUnitalRingHom.id instance : Zero (α →ₙ+* β) := ⟨{ toFun := 0, map_mul' := fun _ _ => (mul_zero (0 : β)).symm, map_zero' := rfl, map_add' := fun _ _ => (add_zero (0 : β)).symm }⟩ instance : Inhabited (α →ₙ+* β) := ⟨0⟩ @[simp] theorem coe_zero : ⇑(0 : α →ₙ+* β) = 0 := rfl #align non_unital_ring_hom.coe_zero NonUnitalRingHom.coe_zero @[simp] theorem zero_apply (x : α) : (0 : α →ₙ+* β) x = 0 := rfl #align non_unital_ring_hom.zero_apply NonUnitalRingHom.zero_apply @[simp] theorem id_apply (x : α) : NonUnitalRingHom.id α x = x := rfl #align non_unital_ring_hom.id_apply NonUnitalRingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (NonUnitalRingHom.id α : α →+ α) = AddMonoidHom.id α := rfl #align non_unital_ring_hom.coe_add_monoid_hom_id NonUnitalRingHom.coe_addMonoidHom_id @[simp] theorem coe_mulHom_id : (NonUnitalRingHom.id α : α →ₙ* α) = MulHom.id α := rfl #align non_unital_ring_hom.coe_mul_hom_id NonUnitalRingHom.coe_mulHom_id variable [NonUnitalNonAssocSemiring γ] /-- Composition of non-unital ring homomorphisms is a non-unital ring homomorphism. -/ def comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : α →ₙ+* γ := { g.toMulHom.comp f.toMulHom, g.toAddMonoidHom.comp f.toAddMonoidHom with } #align non_unital_ring_hom.comp NonUnitalRingHom.comp /-- Composition of non-unital ring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonUnitalNonAssocSemiring δ} (f : α →ₙ+* β) (g : β →ₙ+* γ) (h : γ →ₙ+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align non_unital_ring_hom.comp_assoc NonUnitalRingHom.comp_assoc @[simp] theorem coe_comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : ⇑(g.comp f) = g ∘ f := rfl #align non_unital_ring_hom.coe_comp NonUnitalRingHom.coe_comp @[simp] theorem comp_apply (g : β →ₙ+* γ) (f : α →ₙ+* β) (x : α) : g.comp f x = g (f x) := rfl #align non_unital_ring_hom.comp_apply NonUnitalRingHom.comp_apply variable (g : β →ₙ+* γ) (f : α →ₙ+* β) @[simp] theorem coe_comp_addMonoidHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : AddMonoidHom.mk ⟨g ∘ f, (g.comp f).map_zero'⟩ (g.comp f).map_add' = (g : β →+ γ).comp f := rfl #align non_unital_ring_hom.coe_comp_add_monoid_hom NonUnitalRingHom.coe_comp_addMonoidHom @[simp] theorem coe_comp_mulHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : MulHom.mk (g ∘ f) (g.comp f).map_mul' = (g : β →ₙ* γ).comp f := rfl #align non_unital_ring_hom.coe_comp_mul_hom NonUnitalRingHom.coe_comp_mulHom @[simp] theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 := by ext simp #align non_unital_ring_hom.comp_zero NonUnitalRingHom.comp_zero @[simp] theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 := by ext rfl #align non_unital_ring_hom.zero_comp NonUnitalRingHom.zero_comp @[simp] theorem comp_id (f : α →ₙ+* β) : f.comp (NonUnitalRingHom.id α) = f := ext fun _ => rfl #align non_unital_ring_hom.comp_id NonUnitalRingHom.comp_id @[simp] theorem id_comp (f : α →ₙ+* β) : (NonUnitalRingHom.id β).comp f = f := ext fun _ => rfl #align non_unital_ring_hom.id_comp NonUnitalRingHom.id_comp instance : MonoidWithZero (α →ₙ+* α) where one := NonUnitalRingHom.id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ zero := 0 mul_zero := comp_zero zero_mul := zero_comp theorem one_def : (1 : α →ₙ+* α) = NonUnitalRingHom.id α := rfl #align non_unital_ring_hom.one_def NonUnitalRingHom.one_def @[simp] theorem coe_one : ⇑(1 : α →ₙ+* α) = id := rfl #align non_unital_ring_hom.coe_one NonUnitalRingHom.coe_one theorem mul_def (f g : α →ₙ+* α) : f * g = f.comp g := rfl #align non_unital_ring_hom.mul_def NonUnitalRingHom.mul_def @[simp] theorem coe_mul (f g : α →ₙ+* α) : ⇑(f * g) = f ∘ g := rfl #align non_unital_ring_hom.coe_mul NonUnitalRingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →ₙ+* γ} {f : α →ₙ+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_right NonUnitalRingHom.cancel_right @[simp] theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_left NonUnitalRingHom.cancel_left end NonUnitalRingHom /-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too. This extends from both `MonoidHom` and `MonoidWithZeroHom` in order to put the fields in a sensible order, even though `MonoidWithZeroHom` already extends `MonoidHom`. -/ structure RingHom (α : Type*) (β : Type*) [NonAssocSemiring α] [NonAssocSemiring β] extends α →* β, α →+ β, α →ₙ+* β, α →*₀ β #align ring_hom RingHom /-- `α →+* β` denotes the type of ring homomorphisms from `α` to `β`. -/ infixr:25 " →+* " => RingHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid with zero homomorphism `α →*₀ β`. The `simp`-normal form is `(f : α →*₀ β)`. -/ add_decl_doc RingHom.toMonoidWithZeroHom #align ring_hom.to_monoid_with_zero_hom RingHom.toMonoidWithZeroHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid homomorphism `α →* β`. The `simp`-normal form is `(f : α →* β)`. -/ add_decl_doc RingHom.toMonoidHom #align ring_hom.to_monoid_hom RingHom.toMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc RingHom.toAddMonoidHom #align ring_hom.to_add_monoid_hom RingHom.toAddMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a non-unital ring homomorphism `α →ₙ+* β`. The `simp`-normal form is `(f : α →ₙ+* β)`. -/ add_decl_doc RingHom.toNonUnitalRingHom #align ring_hom.to_non_unital_ring_hom RingHom.toNonUnitalRingHom section RingHomClass /-- `RingHomClass F α β` states that `F` is a type of (semi)ring homomorphisms. You should extend this class when you extend `RingHom`. This extends from both `MonoidHomClass` and `MonoidWithZeroHomClass` in order to put the fields in a sensible order, even though `MonoidWithZeroHomClass` already extends `MonoidHomClass`. -/ class RingHomClass (F : Type*) (α β : outParam (Type*)) [NonAssocSemiring α] [NonAssocSemiring β] extends MonoidHomClass F α β, AddMonoidHomClass F α β, MonoidWithZeroHomClass F α β #align ring_hom_class RingHomClass set_option linter.deprecated false in /-- Ring homomorphisms preserve `bit1`. -/ @[simp] lemma map_bit1 [NonAssocSemiring α] [NonAssocSemiring β] [RingHomClass F α β] (f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) := by simp [bit1] #align map_bit1 map_bit1 -- Porting note: marked `{}` rather than `[]` to prevent dangerous instances variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} [RingHomClass F α β] /-- Turn an element of a type `F` satisfying `RingHomClass F α β` into an actual `RingHom`. This is declared as the default coercion from `F` to `α →+* β`. -/ @[coe] def RingHomClass.toRingHom (f : F) : α →+* β := { (f : α →* β), (f : α →+ β) with } /-- Any type satisfying `RingHomClass` can be cast into `RingHom` via `RingHomClass.toRingHom`. -/ instance : CoeTC F (α →+* β) := ⟨RingHomClass.toRingHom⟩ instance (priority := 100) RingHomClass.toNonUnitalRingHomClass : NonUnitalRingHomClass F α β := { ‹RingHomClass F α β› with } #align ring_hom_class.to_non_unital_ring_hom_class RingHomClass.toNonUnitalRingHomClass end RingHomClass namespace RingHom section coe /-! Throughout this section, some `Semiring` arguments are specified with `{}` instead of `[]`. See note [implicit instance arguments]. -/ variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} instance instRingHomClass : RingHomClass (α →+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := RingHom.map_add' map_zero := RingHom.map_zero' map_mul f := f.map_mul' map_one f := f.map_one' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -- -/ -- instance : CoeFun (α →+* β) fun _ => α → β := -- ⟨RingHom.toFun⟩ initialize_simps_projections RingHom (toFun → apply) -- Porting note: is this lemma still needed in Lean4? -- Porting note: because `f.toFun` really means `f.toMonoidHom.toOneHom.toFun` and -- `toMonoidHom_eq_coe` wants to simplify `f.toMonoidHom` to `(↑f : M →* N)`, this can't -- be a simp lemma anymore -- @[simp] theorem toFun_eq_coe (f : α →+* β) : f.toFun = f := rfl #align ring_hom.to_fun_eq_coe RingHom.toFun_eq_coe @[simp] theorem coe_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α → β) = f := rfl #align ring_hom.coe_mk RingHom.coe_mk @[simp] theorem coe_coe {F : Type*} [RingHomClass F α β] (f : F) : ((f : α →+* β) : α → β) = f := rfl #align ring_hom.coe_coe RingHom.coe_coe attribute [coe] RingHom.toMonoidHom instance coeToMonoidHom : Coe (α →+* β) (α →* β) := ⟨RingHom.toMonoidHom⟩ #align ring_hom.has_coe_monoid_hom RingHom.coeToMonoidHom -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_monoid_hom @[simp] theorem toMonoidHom_eq_coe (f : α →+* β) : f.toMonoidHom = f := rfl #align ring_hom.to_monoid_hom_eq_coe RingHom.toMonoidHom_eq_coe -- Porting note: this can't be a simp lemma anymore -- @[simp] theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f := by rfl #align ring_hom.to_monoid_with_zero_hom_eq_coe RingHom.toMonoidWithZeroHom_eq_coe @[simp] theorem coe_monoidHom_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α →* β) = f := rfl #align ring_hom.coe_monoid_hom_mk RingHom.coe_monoidHom_mk -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_add_monoid_hom @[simp] theorem toAddMonoidHom_eq_coe (f : α →+* β) : f.toAddMonoidHom = f := rfl #align ring_hom.to_add_monoid_hom_eq_coe RingHom.toAddMonoidHom_eq_coe @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃ h₄) : ((⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨⟨f, h₃⟩, h₄⟩ := rfl #align ring_hom.coe_add_monoid_hom_mk RingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ def copy (f : α →+* β) (f' : α → β) (h : f' = f) : α →+* β := { f.toMonoidWithZeroHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align ring_hom.copy RingHom.copy @[simp] theorem coe_copy (f : α →+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align ring_hom.coe_copy RingHom.coe_copy theorem copy_eq (f : α →+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align ring_hom.copy_eq RingHom.copy_eq end coe section variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} (f : α →+* β) {x y : α} theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x := FunLike.congr_fun h x #align ring_hom.congr_fun RingHom.congr_fun theorem congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y := FunLike.congr_arg f h #align ring_hom.congr_arg RingHom.congr_arg theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g := FunLike.coe_injective h #align ring_hom.coe_inj RingHom.coe_inj @[ext] theorem ext ⦃f g : α →+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align ring_hom.ext RingHom.ext theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align ring_hom.ext_iff RingHom.ext_iff @[simp] theorem mk_coe (f : α →+* β) (h₁ h₂ h₃ h₄) : RingHom.mk ⟨⟨f, h₁⟩, h₂⟩ h₃ h₄ = f := ext fun _ => rfl #align ring_hom.mk_coe RingHom.mk_coe theorem coe_addMonoidHom_injective : Injective (fun f : α →+* β => (f : α →+ β)) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align ring_hom.coe_add_monoid_hom_injective RingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_monoidHom_injective : Injective (fun f : α →+* β => (f : α →* β)) := fun _ _ h => ext <| MonoidHom.congr_fun h #align ring_hom.coe_monoid_hom_injective RingHom.coe_monoidHom_injective /-- Ring homomorphisms map zero to zero. -/ protected theorem map_zero (f : α →+* β) : f 0 = 0 := map_zero f #align ring_hom.map_zero RingHom.map_zero /-- Ring homomorphisms map one to one. -/ protected theorem map_one (f : α →+* β) : f 1 = 1 := map_one f #align ring_hom.map_one RingHom.map_one /-- Ring homomorphisms preserve addition. -/ protected theorem map_add (f : α →+* β) : ∀ a b, f (a + b) = f a + f b := map_add f #align ring_hom.map_add RingHom.map_add /-- Ring homomorphisms preserve multiplication. -/ protected theorem map_mul (f : α →+* β) : ∀ a b, f (a * b) = f a * f b := map_mul f #align ring_hom.map_mul RingHom.map_mul @[simp] theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 0 1) = ite p 0 1 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_zero_one RingHom.map_ite_zero_one @[simp] theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 1 0) = ite p 1 0 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_one_zero RingHom.map_ite_one_zero /-- `f : α →+* β` has a trivial codomain iff `f 1 = 0`. -/ theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 := by rw [map_one, eq_comm] #align ring_hom.codomain_trivial_iff_map_one_eq_zero RingHom.codomain_trivial_iff_map_one_eq_zero /-- `f : α →+* β` has a trivial codomain iff it has a trivial range. -/ theorem codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 := f.codomain_trivial_iff_map_one_eq_zero.trans ⟨fun h x => by rw [← mul_one x, map_mul, h, mul_zero], fun h => h 1⟩ #align ring_hom.codomain_trivial_iff_range_trivial RingHom.codomain_trivial_iff_range_trivial /-- `f : α →+* β` doesn't map `1` to `0` if `β` is nontrivial -/ theorem map_one_ne_zero [Nontrivial β] : f 1 ≠ 0 := mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one #align ring_hom.map_one_ne_zero RingHom.map_one_ne_zero /-- If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. -/ theorem domain_nontrivial [Nontrivial β] : Nontrivial α := ⟨⟨1, 0, mt (fun h => show f 1 = 0 by rw [h, map_zero]) f.map_one_ne_zero⟩⟩ #align ring_hom.domain_nontrivial RingHom.domain_nontrivial theorem codomain_trivial (f : α →+* β) [h : Subsingleton α] : Subsingleton β := (subsingleton_or_nontrivial β).resolve_right fun _ => not_nontrivial_iff_subsingleton.mpr h f.domain_nontrivial #align ring_hom.codomain_trivial RingHom.codomain_trivial end /-- Ring homomorphisms preserve additive inverse. -/ protected theorem map_neg [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x : α) : f (-x) = -f x := map_neg f x #align ring_hom.map_neg RingHom.map_neg /-- Ring homomorphisms preserve subtraction. -/ protected theorem map_sub [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x y : α) : f (x - y) = f x - f y := map_sub f x y #align ring_hom.map_sub RingHom.map_sub /-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/ def mk' [NonAssocSemiring α] [NonAssocRing β] (f : α →* β) (map_add : ∀ a b, f (a + b) = f a + f b) : α →+* β := { AddMonoidHom.mk' f map_add, f with } #align ring_hom.mk' RingHom.mk' variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} /-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;> intros <;> rfl #align ring_hom.id RingHom.id instance : Inhabited (α →+* α) := ⟨id α⟩ @[simp] theorem id_apply (x : α) : RingHom.id α x = x := rfl #align ring_hom.id_apply RingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (id α : α →+ α) = AddMonoidHom.id α := rfl #align ring_hom.coe_add_monoid_hom_id RingHom.coe_addMonoidHom_id @[simp] theorem coe_monoidHom_id : (id α : α →* α) = MonoidHom.id α := rfl #align ring_hom.coe_monoid_hom_id RingHom.coe_monoidHom_id variable {_ : NonAssocSemiring γ} /-- Composition of ring homomorphisms is a ring homomorphism. -/ def comp (g : β →+* γ) (f : α →+* β) : α →+* γ := { g.toNonUnitalRingHom.comp f.toNonUnitalRingHom with toFun := g ∘ f, map_one' := by simp } #align ring_hom.comp RingHom.comp /-- Composition of semiring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonAssocSemiring δ} (f : α →+* β) (g : β →+* γ) (h : γ →+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align ring_hom.comp_assoc RingHom.comp_assoc @[simp] theorem coe_comp (hnp : β →+* γ) (hmn : α →+* β) : (hnp.comp hmn : α → γ) = hnp ∘ hmn := rfl #align ring_hom.coe_comp RingHom.coe_comp theorem comp_apply (hnp : β →+* γ) (hmn : α →+* β) (x : α) : (hnp.comp hmn : α → γ) x = hnp (hmn x) := rfl #align ring_hom.comp_apply RingHom.comp_apply @[simp] theorem comp_id (f : α →+* β) : f.comp (id α) = f := ext fun _ => rfl #align ring_hom.comp_id RingHom.comp_id @[simp] theorem id_comp (f : α →+* β) : (id β).comp f = f := ext fun _ => rfl #align ring_hom.id_comp RingHom.id_comp instance : Monoid (α →+* α) where one := id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ theorem one_def : (1 : α →+* α) = id α := rfl #align ring_hom.one_def RingHom.one_def theorem mul_def (f g : α →+* α) : f * g = f.comp g := rfl #align ring_hom.mul_def RingHom.mul_def @[simp] theorem coe_one : ⇑(1 : α →+* α) = _root_.id := rfl #align ring_hom.coe_one RingHom.coe_one @[simp] theorem coe_mul (f g : α →+* α) : ⇑(f * g) = f ∘ g := rfl #align ring_hom.coe_mul RingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →+* γ} {f : α →+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => RingHom.ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align ring_hom.cancel_right RingHom.cancel_right @[simp] theorem cancel_left {g : β →+* γ} {f₁ f₂ : α →+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => RingHom.ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align ring_hom.cancel_left RingHom.cancel_left end RingHom namespace AddMonoidHom variable [CommRing α] [IsDomain α] [CommRing β] (f : β →+ α) -- Porting note: there's some disagreement over the naming scheme here. -- This could perhaps be `mkRingHom_of_mul_self_of_two_ne_zero`. -- See https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/naming.20conventions/near/315558410 /-- Make a ring homomorphism from an additive group homomorphism from a commutative ring to an integral domain that commutes with self multiplication, assumes that two is nonzero and `1` is sent to `1`. -/ def mkRingHomOfMulSelfOfTwoNeZero (h : ∀ x, f (x * x) = f x * f x) (h_two : (2 : α) ≠ 0) (h_one : f 1 = 1) : β →+* α := { f with map_one' := h_one, map_mul' := fun x y => by have hxy := h (x + y) rw [mul_add, add_mul, add_mul, f.map_add, f.map_add, f.map_add, f.map_add, h x, h y, add_mul, mul_add, mul_add, ← sub_eq_zero, add_comm (f x * f x + f (y * x)), ← sub_sub, ← sub_sub, ← sub_sub, mul_comm y x, mul_comm (f y) (f x)] at hxy simp only [add_assoc, add_sub_assoc, add_sub_cancel'_right] at hxy rw [sub_sub, ← two_mul, ← add_sub_assoc, ← two_mul, ← mul_sub, mul_eq_zero (M₀ := α), sub_eq_zero, or_iff_not_imp_left] at hxy

exact hxy h_two

/-- Make a ring homomorphism from an additive group homomorphism from a commutative ring to an integral domain that commutes with self multiplication, assumes that two is nonzero and `1` is sent to `1`. -/ def mkRingHomOfMulSelfOfTwoNeZero (h : ∀ x, f (x * x) = f x * f x) (h_two : (2 : α) ≠ 0) (h_one : f 1 = 1) : β →+* α := { f with map_one' := h_one, map_mul' := fun x y => by have hxy := h (x + y) rw [mul_add, add_mul, add_mul, f.map_add, f.map_add, f.map_add, f.map_add, h x, h y, add_mul, mul_add, mul_add, ← sub_eq_zero, add_comm (f x * f x + f (y * x)), ← sub_sub, ← sub_sub, ← sub_sub, mul_comm y x, mul_comm (f y) (f x)] at hxy simp only [add_assoc, add_sub_assoc, add_sub_cancel'_right] at hxy rw [sub_sub, ← two_mul, ← add_sub_assoc, ← two_mul, ← mul_sub, mul_eq_zero (M₀ := α), sub_eq_zero, or_iff_not_imp_left] at hxy

Mathlib.Algebra.Ring.Hom.Defs.733_0.KyHvVYrIs9pW9ZQ

/-- Make a ring homomorphism from an additive group homomorphism from a commutative ring to an integral domain that commutes with self multiplication, assumes that two is nonzero and `1` is sent to `1`. -/ def mkRingHomOfMulSelfOfTwoNeZero (h : ∀ x, f (x * x) = f x * f x) (h_two : (2 : α) ≠ 0) (h_one : f 1 = 1) : β →+* α

Mathlib_Algebra_Ring_Hom_Defs

F : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : CommRing α inst✝¹ : IsDomain α inst✝ : CommRing β f : β →+ α h : ∀ (x : β), f (x * x) = f x * f x h_two : 2 ≠ 0 h_one : f 1 = 1 ⊢ ↑(mkRingHomOfMulSelfOfTwoNeZero f h h_two h_one) = f

/- Copyright (c) 2019 Amelia Livingston. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Amelia Livingston, Jireh Loreaux -/ import Mathlib.Algebra.Ring.Defs import Mathlib.Algebra.Ring.Basic import Mathlib.Data.Pi.Algebra #align_import algebra.hom.ring from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9" /-! # Homomorphisms of semirings and rings This file defines bundled homomorphisms of (non-unital) semirings and rings. As with monoid and groups, we use the same structure `RingHom a β`, a.k.a. `α →+* β`, for both types of homomorphisms. ## Main definitions * `NonUnitalRingHom`: Non-unital (semi)ring homomorphisms. Additive monoid homomorphism which preserve multiplication. * `RingHom`: (Semi)ring homomorphisms. Monoid homomorphisms which are also additive monoid homomorphism. ## Notations * `→ₙ+*`: Non-unital (semi)ring homs * `→+*`: (Semi)ring homs ## Implementation notes * There's a coercion from bundled homs to fun, and the canonical notation is to use the bundled hom as a function via this coercion. * There is no `SemiringHom` -- the idea is that `RingHom` is used. The constructor for a `RingHom` between semirings needs a proof of `map_zero`, `map_one` and `map_add` as well as `map_mul`; a separate constructor `RingHom.mk'` will construct ring homs between rings from monoid homs given only a proof that addition is preserved. ## Tags `RingHom`, `SemiringHom` -/ open Function variable {F α β γ : Type*} /-- Bundled non-unital semiring homomorphisms `α →ₙ+* β`; use this for bundled non-unital ring homomorphisms too. When possible, instead of parametrizing results over `(f : α →ₙ+* β)`, you should parametrize over `(F : Type*) [NonUnitalRingHomClass F α β] (f : F)`. When you extend this structure, make sure to extend `NonUnitalRingHomClass`. -/ structure NonUnitalRingHom (α β : Type*) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends α →ₙ* β, α →+ β #align non_unital_ring_hom NonUnitalRingHom /-- `α →ₙ+* β` denotes the type of non-unital ring homomorphisms from `α` to `β`. -/ infixr:25 " →ₙ+* " => NonUnitalRingHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as a semigroup homomorphism `α →ₙ* β`. The `simp`-normal form is `(f : α →ₙ* β)`. -/ add_decl_doc NonUnitalRingHom.toMulHom #align non_unital_ring_hom.to_mul_hom NonUnitalRingHom.toMulHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc NonUnitalRingHom.toAddMonoidHom #align non_unital_ring_hom.to_add_monoid_hom NonUnitalRingHom.toAddMonoidHom section NonUnitalRingHomClass /-- `NonUnitalRingHomClass F α β` states that `F` is a type of non-unital (semi)ring homomorphisms. You should extend this class when you extend `NonUnitalRingHom`. -/ class NonUnitalRingHomClass (F : Type*) (α β : outParam (Type*)) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends MulHomClass F α β, AddMonoidHomClass F α β #align non_unital_ring_hom_class NonUnitalRingHomClass variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] [NonUnitalRingHomClass F α β] /-- Turn an element of a type `F` satisfying `NonUnitalRingHomClass F α β` into an actual `NonUnitalRingHom`. This is declared as the default coercion from `F` to `α →ₙ+* β`. -/ @[coe] def NonUnitalRingHomClass.toNonUnitalRingHom (f : F) : α →ₙ+* β := { (f : α →ₙ* β), (f : α →+ β) with } /-- Any type satisfying `NonUnitalRingHomClass` can be cast into `NonUnitalRingHom` via `NonUnitalRingHomClass.toNonUnitalRingHom`. -/ instance : CoeTC F (α →ₙ+* β) := ⟨NonUnitalRingHomClass.toNonUnitalRingHom⟩ end NonUnitalRingHomClass namespace NonUnitalRingHom section coe variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] instance : NonUnitalRingHomClass (α →ₙ+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := NonUnitalRingHom.map_add' map_zero := NonUnitalRingHom.map_zero' map_mul f := f.map_mul' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -/ -- instance : CoeFun (α →ₙ+* β) fun _ => α → β := -- ⟨fun f => f.toFun⟩ -- Porting note: removed due to new `coe` in Lean4 #noalign non_unital_ring_hom.to_fun_eq_coe #noalign non_unital_ring_hom.coe_mk #noalign non_unital_ring_hom.coe_coe initialize_simps_projections NonUnitalRingHom (toFun → apply) @[simp] theorem coe_toMulHom (f : α →ₙ+* β) : ⇑f.toMulHom = f := rfl #align non_unital_ring_hom.coe_to_mul_hom NonUnitalRingHom.coe_toMulHom @[simp] theorem coe_mulHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →ₙ* β) = ⟨f, h₁⟩ := rfl #align non_unital_ring_hom.coe_mul_hom_mk NonUnitalRingHom.coe_mulHom_mk theorem coe_toAddMonoidHom (f : α →ₙ+* β) : ⇑f.toAddMonoidHom = f := rfl #align non_unital_ring_hom.coe_to_add_monoid_hom NonUnitalRingHom.coe_toAddMonoidHom @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →+ β) = ⟨⟨f, h₂⟩, h₃⟩ := rfl #align non_unital_ring_hom.coe_add_monoid_hom_mk NonUnitalRingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : α →ₙ+* β := { f.toMulHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align non_unital_ring_hom.copy NonUnitalRingHom.copy @[simp] theorem coe_copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align non_unital_ring_hom.coe_copy NonUnitalRingHom.coe_copy theorem copy_eq (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align non_unital_ring_hom.copy_eq NonUnitalRingHom.copy_eq end coe section variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] variable (f : α →ₙ+* β) {x y : α} @[ext] theorem ext ⦃f g : α →ₙ+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align non_unital_ring_hom.ext NonUnitalRingHom.ext theorem ext_iff {f g : α →ₙ+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align non_unital_ring_hom.ext_iff NonUnitalRingHom.ext_iff @[simp] theorem mk_coe (f : α →ₙ+* β) (h₁ h₂ h₃) : NonUnitalRingHom.mk (MulHom.mk f h₁) h₂ h₃ = f := ext fun _ => rfl #align non_unital_ring_hom.mk_coe NonUnitalRingHom.mk_coe theorem coe_addMonoidHom_injective : Injective fun f : α →ₙ+* β => (f : α →+ β) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align non_unital_ring_hom.coe_add_monoid_hom_injective NonUnitalRingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_mulHom_injective : Injective fun f : α →ₙ+* β => (f : α →ₙ* β) := fun _ _ h => ext <| MulHom.congr_fun h #align non_unital_ring_hom.coe_mul_hom_injective NonUnitalRingHom.coe_mulHom_injective end variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/ protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by refine' { toFun := id.. } <;> intros <;> rfl #align non_unital_ring_hom.id NonUnitalRingHom.id instance : Zero (α →ₙ+* β) := ⟨{ toFun := 0, map_mul' := fun _ _ => (mul_zero (0 : β)).symm, map_zero' := rfl, map_add' := fun _ _ => (add_zero (0 : β)).symm }⟩ instance : Inhabited (α →ₙ+* β) := ⟨0⟩ @[simp] theorem coe_zero : ⇑(0 : α →ₙ+* β) = 0 := rfl #align non_unital_ring_hom.coe_zero NonUnitalRingHom.coe_zero @[simp] theorem zero_apply (x : α) : (0 : α →ₙ+* β) x = 0 := rfl #align non_unital_ring_hom.zero_apply NonUnitalRingHom.zero_apply @[simp] theorem id_apply (x : α) : NonUnitalRingHom.id α x = x := rfl #align non_unital_ring_hom.id_apply NonUnitalRingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (NonUnitalRingHom.id α : α →+ α) = AddMonoidHom.id α := rfl #align non_unital_ring_hom.coe_add_monoid_hom_id NonUnitalRingHom.coe_addMonoidHom_id @[simp] theorem coe_mulHom_id : (NonUnitalRingHom.id α : α →ₙ* α) = MulHom.id α := rfl #align non_unital_ring_hom.coe_mul_hom_id NonUnitalRingHom.coe_mulHom_id variable [NonUnitalNonAssocSemiring γ] /-- Composition of non-unital ring homomorphisms is a non-unital ring homomorphism. -/ def comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : α →ₙ+* γ := { g.toMulHom.comp f.toMulHom, g.toAddMonoidHom.comp f.toAddMonoidHom with } #align non_unital_ring_hom.comp NonUnitalRingHom.comp /-- Composition of non-unital ring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonUnitalNonAssocSemiring δ} (f : α →ₙ+* β) (g : β →ₙ+* γ) (h : γ →ₙ+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align non_unital_ring_hom.comp_assoc NonUnitalRingHom.comp_assoc @[simp] theorem coe_comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : ⇑(g.comp f) = g ∘ f := rfl #align non_unital_ring_hom.coe_comp NonUnitalRingHom.coe_comp @[simp] theorem comp_apply (g : β →ₙ+* γ) (f : α →ₙ+* β) (x : α) : g.comp f x = g (f x) := rfl #align non_unital_ring_hom.comp_apply NonUnitalRingHom.comp_apply variable (g : β →ₙ+* γ) (f : α →ₙ+* β) @[simp] theorem coe_comp_addMonoidHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : AddMonoidHom.mk ⟨g ∘ f, (g.comp f).map_zero'⟩ (g.comp f).map_add' = (g : β →+ γ).comp f := rfl #align non_unital_ring_hom.coe_comp_add_monoid_hom NonUnitalRingHom.coe_comp_addMonoidHom @[simp] theorem coe_comp_mulHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : MulHom.mk (g ∘ f) (g.comp f).map_mul' = (g : β →ₙ* γ).comp f := rfl #align non_unital_ring_hom.coe_comp_mul_hom NonUnitalRingHom.coe_comp_mulHom @[simp] theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 := by ext simp #align non_unital_ring_hom.comp_zero NonUnitalRingHom.comp_zero @[simp] theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 := by ext rfl #align non_unital_ring_hom.zero_comp NonUnitalRingHom.zero_comp @[simp] theorem comp_id (f : α →ₙ+* β) : f.comp (NonUnitalRingHom.id α) = f := ext fun _ => rfl #align non_unital_ring_hom.comp_id NonUnitalRingHom.comp_id @[simp] theorem id_comp (f : α →ₙ+* β) : (NonUnitalRingHom.id β).comp f = f := ext fun _ => rfl #align non_unital_ring_hom.id_comp NonUnitalRingHom.id_comp instance : MonoidWithZero (α →ₙ+* α) where one := NonUnitalRingHom.id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ zero := 0 mul_zero := comp_zero zero_mul := zero_comp theorem one_def : (1 : α →ₙ+* α) = NonUnitalRingHom.id α := rfl #align non_unital_ring_hom.one_def NonUnitalRingHom.one_def @[simp] theorem coe_one : ⇑(1 : α →ₙ+* α) = id := rfl #align non_unital_ring_hom.coe_one NonUnitalRingHom.coe_one theorem mul_def (f g : α →ₙ+* α) : f * g = f.comp g := rfl #align non_unital_ring_hom.mul_def NonUnitalRingHom.mul_def @[simp] theorem coe_mul (f g : α →ₙ+* α) : ⇑(f * g) = f ∘ g := rfl #align non_unital_ring_hom.coe_mul NonUnitalRingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →ₙ+* γ} {f : α →ₙ+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_right NonUnitalRingHom.cancel_right @[simp] theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_left NonUnitalRingHom.cancel_left end NonUnitalRingHom /-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too. This extends from both `MonoidHom` and `MonoidWithZeroHom` in order to put the fields in a sensible order, even though `MonoidWithZeroHom` already extends `MonoidHom`. -/ structure RingHom (α : Type*) (β : Type*) [NonAssocSemiring α] [NonAssocSemiring β] extends α →* β, α →+ β, α →ₙ+* β, α →*₀ β #align ring_hom RingHom /-- `α →+* β` denotes the type of ring homomorphisms from `α` to `β`. -/ infixr:25 " →+* " => RingHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid with zero homomorphism `α →*₀ β`. The `simp`-normal form is `(f : α →*₀ β)`. -/ add_decl_doc RingHom.toMonoidWithZeroHom #align ring_hom.to_monoid_with_zero_hom RingHom.toMonoidWithZeroHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid homomorphism `α →* β`. The `simp`-normal form is `(f : α →* β)`. -/ add_decl_doc RingHom.toMonoidHom #align ring_hom.to_monoid_hom RingHom.toMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc RingHom.toAddMonoidHom #align ring_hom.to_add_monoid_hom RingHom.toAddMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a non-unital ring homomorphism `α →ₙ+* β`. The `simp`-normal form is `(f : α →ₙ+* β)`. -/ add_decl_doc RingHom.toNonUnitalRingHom #align ring_hom.to_non_unital_ring_hom RingHom.toNonUnitalRingHom section RingHomClass /-- `RingHomClass F α β` states that `F` is a type of (semi)ring homomorphisms. You should extend this class when you extend `RingHom`. This extends from both `MonoidHomClass` and `MonoidWithZeroHomClass` in order to put the fields in a sensible order, even though `MonoidWithZeroHomClass` already extends `MonoidHomClass`. -/ class RingHomClass (F : Type*) (α β : outParam (Type*)) [NonAssocSemiring α] [NonAssocSemiring β] extends MonoidHomClass F α β, AddMonoidHomClass F α β, MonoidWithZeroHomClass F α β #align ring_hom_class RingHomClass set_option linter.deprecated false in /-- Ring homomorphisms preserve `bit1`. -/ @[simp] lemma map_bit1 [NonAssocSemiring α] [NonAssocSemiring β] [RingHomClass F α β] (f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) := by simp [bit1] #align map_bit1 map_bit1 -- Porting note: marked `{}` rather than `[]` to prevent dangerous instances variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} [RingHomClass F α β] /-- Turn an element of a type `F` satisfying `RingHomClass F α β` into an actual `RingHom`. This is declared as the default coercion from `F` to `α →+* β`. -/ @[coe] def RingHomClass.toRingHom (f : F) : α →+* β := { (f : α →* β), (f : α →+ β) with } /-- Any type satisfying `RingHomClass` can be cast into `RingHom` via `RingHomClass.toRingHom`. -/ instance : CoeTC F (α →+* β) := ⟨RingHomClass.toRingHom⟩ instance (priority := 100) RingHomClass.toNonUnitalRingHomClass : NonUnitalRingHomClass F α β := { ‹RingHomClass F α β› with } #align ring_hom_class.to_non_unital_ring_hom_class RingHomClass.toNonUnitalRingHomClass end RingHomClass namespace RingHom section coe /-! Throughout this section, some `Semiring` arguments are specified with `{}` instead of `[]`. See note [implicit instance arguments]. -/ variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} instance instRingHomClass : RingHomClass (α →+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := RingHom.map_add' map_zero := RingHom.map_zero' map_mul f := f.map_mul' map_one f := f.map_one' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -- -/ -- instance : CoeFun (α →+* β) fun _ => α → β := -- ⟨RingHom.toFun⟩ initialize_simps_projections RingHom (toFun → apply) -- Porting note: is this lemma still needed in Lean4? -- Porting note: because `f.toFun` really means `f.toMonoidHom.toOneHom.toFun` and -- `toMonoidHom_eq_coe` wants to simplify `f.toMonoidHom` to `(↑f : M →* N)`, this can't -- be a simp lemma anymore -- @[simp] theorem toFun_eq_coe (f : α →+* β) : f.toFun = f := rfl #align ring_hom.to_fun_eq_coe RingHom.toFun_eq_coe @[simp] theorem coe_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α → β) = f := rfl #align ring_hom.coe_mk RingHom.coe_mk @[simp] theorem coe_coe {F : Type*} [RingHomClass F α β] (f : F) : ((f : α →+* β) : α → β) = f := rfl #align ring_hom.coe_coe RingHom.coe_coe attribute [coe] RingHom.toMonoidHom instance coeToMonoidHom : Coe (α →+* β) (α →* β) := ⟨RingHom.toMonoidHom⟩ #align ring_hom.has_coe_monoid_hom RingHom.coeToMonoidHom -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_monoid_hom @[simp] theorem toMonoidHom_eq_coe (f : α →+* β) : f.toMonoidHom = f := rfl #align ring_hom.to_monoid_hom_eq_coe RingHom.toMonoidHom_eq_coe -- Porting note: this can't be a simp lemma anymore -- @[simp] theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f := by rfl #align ring_hom.to_monoid_with_zero_hom_eq_coe RingHom.toMonoidWithZeroHom_eq_coe @[simp] theorem coe_monoidHom_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α →* β) = f := rfl #align ring_hom.coe_monoid_hom_mk RingHom.coe_monoidHom_mk -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_add_monoid_hom @[simp] theorem toAddMonoidHom_eq_coe (f : α →+* β) : f.toAddMonoidHom = f := rfl #align ring_hom.to_add_monoid_hom_eq_coe RingHom.toAddMonoidHom_eq_coe @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃ h₄) : ((⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨⟨f, h₃⟩, h₄⟩ := rfl #align ring_hom.coe_add_monoid_hom_mk RingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ def copy (f : α →+* β) (f' : α → β) (h : f' = f) : α →+* β := { f.toMonoidWithZeroHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align ring_hom.copy RingHom.copy @[simp] theorem coe_copy (f : α →+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align ring_hom.coe_copy RingHom.coe_copy theorem copy_eq (f : α →+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align ring_hom.copy_eq RingHom.copy_eq end coe section variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} (f : α →+* β) {x y : α} theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x := FunLike.congr_fun h x #align ring_hom.congr_fun RingHom.congr_fun theorem congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y := FunLike.congr_arg f h #align ring_hom.congr_arg RingHom.congr_arg theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g := FunLike.coe_injective h #align ring_hom.coe_inj RingHom.coe_inj @[ext] theorem ext ⦃f g : α →+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align ring_hom.ext RingHom.ext theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align ring_hom.ext_iff RingHom.ext_iff @[simp] theorem mk_coe (f : α →+* β) (h₁ h₂ h₃ h₄) : RingHom.mk ⟨⟨f, h₁⟩, h₂⟩ h₃ h₄ = f := ext fun _ => rfl #align ring_hom.mk_coe RingHom.mk_coe theorem coe_addMonoidHom_injective : Injective (fun f : α →+* β => (f : α →+ β)) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align ring_hom.coe_add_monoid_hom_injective RingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_monoidHom_injective : Injective (fun f : α →+* β => (f : α →* β)) := fun _ _ h => ext <| MonoidHom.congr_fun h #align ring_hom.coe_monoid_hom_injective RingHom.coe_monoidHom_injective /-- Ring homomorphisms map zero to zero. -/ protected theorem map_zero (f : α →+* β) : f 0 = 0 := map_zero f #align ring_hom.map_zero RingHom.map_zero /-- Ring homomorphisms map one to one. -/ protected theorem map_one (f : α →+* β) : f 1 = 1 := map_one f #align ring_hom.map_one RingHom.map_one /-- Ring homomorphisms preserve addition. -/ protected theorem map_add (f : α →+* β) : ∀ a b, f (a + b) = f a + f b := map_add f #align ring_hom.map_add RingHom.map_add /-- Ring homomorphisms preserve multiplication. -/ protected theorem map_mul (f : α →+* β) : ∀ a b, f (a * b) = f a * f b := map_mul f #align ring_hom.map_mul RingHom.map_mul @[simp] theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 0 1) = ite p 0 1 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_zero_one RingHom.map_ite_zero_one @[simp] theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 1 0) = ite p 1 0 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_one_zero RingHom.map_ite_one_zero /-- `f : α →+* β` has a trivial codomain iff `f 1 = 0`. -/ theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 := by rw [map_one, eq_comm] #align ring_hom.codomain_trivial_iff_map_one_eq_zero RingHom.codomain_trivial_iff_map_one_eq_zero /-- `f : α →+* β` has a trivial codomain iff it has a trivial range. -/ theorem codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 := f.codomain_trivial_iff_map_one_eq_zero.trans ⟨fun h x => by rw [← mul_one x, map_mul, h, mul_zero], fun h => h 1⟩ #align ring_hom.codomain_trivial_iff_range_trivial RingHom.codomain_trivial_iff_range_trivial /-- `f : α →+* β` doesn't map `1` to `0` if `β` is nontrivial -/ theorem map_one_ne_zero [Nontrivial β] : f 1 ≠ 0 := mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one #align ring_hom.map_one_ne_zero RingHom.map_one_ne_zero /-- If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. -/ theorem domain_nontrivial [Nontrivial β] : Nontrivial α := ⟨⟨1, 0, mt (fun h => show f 1 = 0 by rw [h, map_zero]) f.map_one_ne_zero⟩⟩ #align ring_hom.domain_nontrivial RingHom.domain_nontrivial theorem codomain_trivial (f : α →+* β) [h : Subsingleton α] : Subsingleton β := (subsingleton_or_nontrivial β).resolve_right fun _ => not_nontrivial_iff_subsingleton.mpr h f.domain_nontrivial #align ring_hom.codomain_trivial RingHom.codomain_trivial end /-- Ring homomorphisms preserve additive inverse. -/ protected theorem map_neg [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x : α) : f (-x) = -f x := map_neg f x #align ring_hom.map_neg RingHom.map_neg /-- Ring homomorphisms preserve subtraction. -/ protected theorem map_sub [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x y : α) : f (x - y) = f x - f y := map_sub f x y #align ring_hom.map_sub RingHom.map_sub /-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/ def mk' [NonAssocSemiring α] [NonAssocRing β] (f : α →* β) (map_add : ∀ a b, f (a + b) = f a + f b) : α →+* β := { AddMonoidHom.mk' f map_add, f with } #align ring_hom.mk' RingHom.mk' variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} /-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;> intros <;> rfl #align ring_hom.id RingHom.id instance : Inhabited (α →+* α) := ⟨id α⟩ @[simp] theorem id_apply (x : α) : RingHom.id α x = x := rfl #align ring_hom.id_apply RingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (id α : α →+ α) = AddMonoidHom.id α := rfl #align ring_hom.coe_add_monoid_hom_id RingHom.coe_addMonoidHom_id @[simp] theorem coe_monoidHom_id : (id α : α →* α) = MonoidHom.id α := rfl #align ring_hom.coe_monoid_hom_id RingHom.coe_monoidHom_id variable {_ : NonAssocSemiring γ} /-- Composition of ring homomorphisms is a ring homomorphism. -/ def comp (g : β →+* γ) (f : α →+* β) : α →+* γ := { g.toNonUnitalRingHom.comp f.toNonUnitalRingHom with toFun := g ∘ f, map_one' := by simp } #align ring_hom.comp RingHom.comp /-- Composition of semiring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonAssocSemiring δ} (f : α →+* β) (g : β →+* γ) (h : γ →+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align ring_hom.comp_assoc RingHom.comp_assoc @[simp] theorem coe_comp (hnp : β →+* γ) (hmn : α →+* β) : (hnp.comp hmn : α → γ) = hnp ∘ hmn := rfl #align ring_hom.coe_comp RingHom.coe_comp theorem comp_apply (hnp : β →+* γ) (hmn : α →+* β) (x : α) : (hnp.comp hmn : α → γ) x = hnp (hmn x) := rfl #align ring_hom.comp_apply RingHom.comp_apply @[simp] theorem comp_id (f : α →+* β) : f.comp (id α) = f := ext fun _ => rfl #align ring_hom.comp_id RingHom.comp_id @[simp] theorem id_comp (f : α →+* β) : (id β).comp f = f := ext fun _ => rfl #align ring_hom.id_comp RingHom.id_comp instance : Monoid (α →+* α) where one := id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ theorem one_def : (1 : α →+* α) = id α := rfl #align ring_hom.one_def RingHom.one_def theorem mul_def (f g : α →+* α) : f * g = f.comp g := rfl #align ring_hom.mul_def RingHom.mul_def @[simp] theorem coe_one : ⇑(1 : α →+* α) = _root_.id := rfl #align ring_hom.coe_one RingHom.coe_one @[simp] theorem coe_mul (f g : α →+* α) : ⇑(f * g) = f ∘ g := rfl #align ring_hom.coe_mul RingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →+* γ} {f : α →+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => RingHom.ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align ring_hom.cancel_right RingHom.cancel_right @[simp] theorem cancel_left {g : β →+* γ} {f₁ f₂ : α →+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => RingHom.ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align ring_hom.cancel_left RingHom.cancel_left end RingHom namespace AddMonoidHom variable [CommRing α] [IsDomain α] [CommRing β] (f : β →+ α) -- Porting note: there's some disagreement over the naming scheme here. -- This could perhaps be `mkRingHom_of_mul_self_of_two_ne_zero`. -- See https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/naming.20conventions/near/315558410 /-- Make a ring homomorphism from an additive group homomorphism from a commutative ring to an integral domain that commutes with self multiplication, assumes that two is nonzero and `1` is sent to `1`. -/ def mkRingHomOfMulSelfOfTwoNeZero (h : ∀ x, f (x * x) = f x * f x) (h_two : (2 : α) ≠ 0) (h_one : f 1 = 1) : β →+* α := { f with map_one' := h_one, map_mul' := fun x y => by have hxy := h (x + y) rw [mul_add, add_mul, add_mul, f.map_add, f.map_add, f.map_add, f.map_add, h x, h y, add_mul, mul_add, mul_add, ← sub_eq_zero, add_comm (f x * f x + f (y * x)), ← sub_sub, ← sub_sub, ← sub_sub, mul_comm y x, mul_comm (f y) (f x)] at hxy simp only [add_assoc, add_sub_assoc, add_sub_cancel'_right] at hxy rw [sub_sub, ← two_mul, ← add_sub_assoc, ← two_mul, ← mul_sub, mul_eq_zero (M₀ := α), sub_eq_zero, or_iff_not_imp_left] at hxy exact hxy h_two } #align add_monoid_hom.mk_ring_hom_of_mul_self_of_two_ne_zero AddMonoidHom.mkRingHomOfMulSelfOfTwoNeZero @[simp] theorem coe_fn_mkRingHomOfMulSelfOfTwoNeZero (h h_two h_one) : (f.mkRingHomOfMulSelfOfTwoNeZero h h_two h_one : β → α) = f := rfl #align add_monoid_hom.coe_fn_mk_ring_hom_of_mul_self_of_two_ne_zero AddMonoidHom.coe_fn_mkRingHomOfMulSelfOfTwoNeZero -- Porting note: `simp` can prove this -- @[simp] theorem coe_addMonoidHom_mkRingHomOfMulSelfOfTwoNeZero (h h_two h_one) : (f.mkRingHomOfMulSelfOfTwoNeZero h h_two h_one : β →+ α) = f := by

ext

theorem coe_addMonoidHom_mkRingHomOfMulSelfOfTwoNeZero (h h_two h_one) : (f.mkRingHomOfMulSelfOfTwoNeZero h h_two h_one : β →+ α) = f := by

Mathlib.Algebra.Ring.Hom.Defs.759_0.KyHvVYrIs9pW9ZQ

theorem coe_addMonoidHom_mkRingHomOfMulSelfOfTwoNeZero (h h_two h_one) : (f.mkRingHomOfMulSelfOfTwoNeZero h h_two h_one : β →+ α) = f

Mathlib_Algebra_Ring_Hom_Defs

case h F : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 inst✝² : CommRing α inst✝¹ : IsDomain α inst✝ : CommRing β f : β →+ α h : ∀ (x : β), f (x * x) = f x * f x h_two : 2 ≠ 0 h_one : f 1 = 1 x✝ : β ⊢ ↑(mkRingHomOfMulSelfOfTwoNeZero f h h_two h_one) x✝ = f x✝

/- Copyright (c) 2019 Amelia Livingston. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Amelia Livingston, Jireh Loreaux -/ import Mathlib.Algebra.Ring.Defs import Mathlib.Algebra.Ring.Basic import Mathlib.Data.Pi.Algebra #align_import algebra.hom.ring from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9" /-! # Homomorphisms of semirings and rings This file defines bundled homomorphisms of (non-unital) semirings and rings. As with monoid and groups, we use the same structure `RingHom a β`, a.k.a. `α →+* β`, for both types of homomorphisms. ## Main definitions * `NonUnitalRingHom`: Non-unital (semi)ring homomorphisms. Additive monoid homomorphism which preserve multiplication. * `RingHom`: (Semi)ring homomorphisms. Monoid homomorphisms which are also additive monoid homomorphism. ## Notations * `→ₙ+*`: Non-unital (semi)ring homs * `→+*`: (Semi)ring homs ## Implementation notes * There's a coercion from bundled homs to fun, and the canonical notation is to use the bundled hom as a function via this coercion. * There is no `SemiringHom` -- the idea is that `RingHom` is used. The constructor for a `RingHom` between semirings needs a proof of `map_zero`, `map_one` and `map_add` as well as `map_mul`; a separate constructor `RingHom.mk'` will construct ring homs between rings from monoid homs given only a proof that addition is preserved. ## Tags `RingHom`, `SemiringHom` -/ open Function variable {F α β γ : Type*} /-- Bundled non-unital semiring homomorphisms `α →ₙ+* β`; use this for bundled non-unital ring homomorphisms too. When possible, instead of parametrizing results over `(f : α →ₙ+* β)`, you should parametrize over `(F : Type*) [NonUnitalRingHomClass F α β] (f : F)`. When you extend this structure, make sure to extend `NonUnitalRingHomClass`. -/ structure NonUnitalRingHom (α β : Type*) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends α →ₙ* β, α →+ β #align non_unital_ring_hom NonUnitalRingHom /-- `α →ₙ+* β` denotes the type of non-unital ring homomorphisms from `α` to `β`. -/ infixr:25 " →ₙ+* " => NonUnitalRingHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as a semigroup homomorphism `α →ₙ* β`. The `simp`-normal form is `(f : α →ₙ* β)`. -/ add_decl_doc NonUnitalRingHom.toMulHom #align non_unital_ring_hom.to_mul_hom NonUnitalRingHom.toMulHom /-- Reinterpret a non-unital ring homomorphism `f : α →ₙ+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc NonUnitalRingHom.toAddMonoidHom #align non_unital_ring_hom.to_add_monoid_hom NonUnitalRingHom.toAddMonoidHom section NonUnitalRingHomClass /-- `NonUnitalRingHomClass F α β` states that `F` is a type of non-unital (semi)ring homomorphisms. You should extend this class when you extend `NonUnitalRingHom`. -/ class NonUnitalRingHomClass (F : Type*) (α β : outParam (Type*)) [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] extends MulHomClass F α β, AddMonoidHomClass F α β #align non_unital_ring_hom_class NonUnitalRingHomClass variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] [NonUnitalRingHomClass F α β] /-- Turn an element of a type `F` satisfying `NonUnitalRingHomClass F α β` into an actual `NonUnitalRingHom`. This is declared as the default coercion from `F` to `α →ₙ+* β`. -/ @[coe] def NonUnitalRingHomClass.toNonUnitalRingHom (f : F) : α →ₙ+* β := { (f : α →ₙ* β), (f : α →+ β) with } /-- Any type satisfying `NonUnitalRingHomClass` can be cast into `NonUnitalRingHom` via `NonUnitalRingHomClass.toNonUnitalRingHom`. -/ instance : CoeTC F (α →ₙ+* β) := ⟨NonUnitalRingHomClass.toNonUnitalRingHom⟩ end NonUnitalRingHomClass namespace NonUnitalRingHom section coe variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] instance : NonUnitalRingHomClass (α →ₙ+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := NonUnitalRingHom.map_add' map_zero := NonUnitalRingHom.map_zero' map_mul f := f.map_mul' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -/ -- instance : CoeFun (α →ₙ+* β) fun _ => α → β := -- ⟨fun f => f.toFun⟩ -- Porting note: removed due to new `coe` in Lean4 #noalign non_unital_ring_hom.to_fun_eq_coe #noalign non_unital_ring_hom.coe_mk #noalign non_unital_ring_hom.coe_coe initialize_simps_projections NonUnitalRingHom (toFun → apply) @[simp] theorem coe_toMulHom (f : α →ₙ+* β) : ⇑f.toMulHom = f := rfl #align non_unital_ring_hom.coe_to_mul_hom NonUnitalRingHom.coe_toMulHom @[simp] theorem coe_mulHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →ₙ* β) = ⟨f, h₁⟩ := rfl #align non_unital_ring_hom.coe_mul_hom_mk NonUnitalRingHom.coe_mulHom_mk theorem coe_toAddMonoidHom (f : α →ₙ+* β) : ⇑f.toAddMonoidHom = f := rfl #align non_unital_ring_hom.coe_to_add_monoid_hom NonUnitalRingHom.coe_toAddMonoidHom @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃) : ((⟨⟨f, h₁⟩, h₂, h₃⟩ : α →ₙ+* β) : α →+ β) = ⟨⟨f, h₂⟩, h₃⟩ := rfl #align non_unital_ring_hom.coe_add_monoid_hom_mk NonUnitalRingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : α →ₙ+* β := { f.toMulHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align non_unital_ring_hom.copy NonUnitalRingHom.copy @[simp] theorem coe_copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align non_unital_ring_hom.coe_copy NonUnitalRingHom.coe_copy theorem copy_eq (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align non_unital_ring_hom.copy_eq NonUnitalRingHom.copy_eq end coe section variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] variable (f : α →ₙ+* β) {x y : α} @[ext] theorem ext ⦃f g : α →ₙ+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align non_unital_ring_hom.ext NonUnitalRingHom.ext theorem ext_iff {f g : α →ₙ+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align non_unital_ring_hom.ext_iff NonUnitalRingHom.ext_iff @[simp] theorem mk_coe (f : α →ₙ+* β) (h₁ h₂ h₃) : NonUnitalRingHom.mk (MulHom.mk f h₁) h₂ h₃ = f := ext fun _ => rfl #align non_unital_ring_hom.mk_coe NonUnitalRingHom.mk_coe theorem coe_addMonoidHom_injective : Injective fun f : α →ₙ+* β => (f : α →+ β) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align non_unital_ring_hom.coe_add_monoid_hom_injective NonUnitalRingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_mulHom_injective : Injective fun f : α →ₙ+* β => (f : α →ₙ* β) := fun _ _ h => ext <| MulHom.congr_fun h #align non_unital_ring_hom.coe_mul_hom_injective NonUnitalRingHom.coe_mulHom_injective end variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] /-- The identity non-unital ring homomorphism from a non-unital semiring to itself. -/ protected def id (α : Type*) [NonUnitalNonAssocSemiring α] : α →ₙ+* α := by refine' { toFun := id.. } <;> intros <;> rfl #align non_unital_ring_hom.id NonUnitalRingHom.id instance : Zero (α →ₙ+* β) := ⟨{ toFun := 0, map_mul' := fun _ _ => (mul_zero (0 : β)).symm, map_zero' := rfl, map_add' := fun _ _ => (add_zero (0 : β)).symm }⟩ instance : Inhabited (α →ₙ+* β) := ⟨0⟩ @[simp] theorem coe_zero : ⇑(0 : α →ₙ+* β) = 0 := rfl #align non_unital_ring_hom.coe_zero NonUnitalRingHom.coe_zero @[simp] theorem zero_apply (x : α) : (0 : α →ₙ+* β) x = 0 := rfl #align non_unital_ring_hom.zero_apply NonUnitalRingHom.zero_apply @[simp] theorem id_apply (x : α) : NonUnitalRingHom.id α x = x := rfl #align non_unital_ring_hom.id_apply NonUnitalRingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (NonUnitalRingHom.id α : α →+ α) = AddMonoidHom.id α := rfl #align non_unital_ring_hom.coe_add_monoid_hom_id NonUnitalRingHom.coe_addMonoidHom_id @[simp] theorem coe_mulHom_id : (NonUnitalRingHom.id α : α →ₙ* α) = MulHom.id α := rfl #align non_unital_ring_hom.coe_mul_hom_id NonUnitalRingHom.coe_mulHom_id variable [NonUnitalNonAssocSemiring γ] /-- Composition of non-unital ring homomorphisms is a non-unital ring homomorphism. -/ def comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : α →ₙ+* γ := { g.toMulHom.comp f.toMulHom, g.toAddMonoidHom.comp f.toAddMonoidHom with } #align non_unital_ring_hom.comp NonUnitalRingHom.comp /-- Composition of non-unital ring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonUnitalNonAssocSemiring δ} (f : α →ₙ+* β) (g : β →ₙ+* γ) (h : γ →ₙ+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align non_unital_ring_hom.comp_assoc NonUnitalRingHom.comp_assoc @[simp] theorem coe_comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : ⇑(g.comp f) = g ∘ f := rfl #align non_unital_ring_hom.coe_comp NonUnitalRingHom.coe_comp @[simp] theorem comp_apply (g : β →ₙ+* γ) (f : α →ₙ+* β) (x : α) : g.comp f x = g (f x) := rfl #align non_unital_ring_hom.comp_apply NonUnitalRingHom.comp_apply variable (g : β →ₙ+* γ) (f : α →ₙ+* β) @[simp] theorem coe_comp_addMonoidHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : AddMonoidHom.mk ⟨g ∘ f, (g.comp f).map_zero'⟩ (g.comp f).map_add' = (g : β →+ γ).comp f := rfl #align non_unital_ring_hom.coe_comp_add_monoid_hom NonUnitalRingHom.coe_comp_addMonoidHom @[simp] theorem coe_comp_mulHom (g : β →ₙ+* γ) (f : α →ₙ+* β) : MulHom.mk (g ∘ f) (g.comp f).map_mul' = (g : β →ₙ* γ).comp f := rfl #align non_unital_ring_hom.coe_comp_mul_hom NonUnitalRingHom.coe_comp_mulHom @[simp] theorem comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 := by ext simp #align non_unital_ring_hom.comp_zero NonUnitalRingHom.comp_zero @[simp] theorem zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 := by ext rfl #align non_unital_ring_hom.zero_comp NonUnitalRingHom.zero_comp @[simp] theorem comp_id (f : α →ₙ+* β) : f.comp (NonUnitalRingHom.id α) = f := ext fun _ => rfl #align non_unital_ring_hom.comp_id NonUnitalRingHom.comp_id @[simp] theorem id_comp (f : α →ₙ+* β) : (NonUnitalRingHom.id β).comp f = f := ext fun _ => rfl #align non_unital_ring_hom.id_comp NonUnitalRingHom.id_comp instance : MonoidWithZero (α →ₙ+* α) where one := NonUnitalRingHom.id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ zero := 0 mul_zero := comp_zero zero_mul := zero_comp theorem one_def : (1 : α →ₙ+* α) = NonUnitalRingHom.id α := rfl #align non_unital_ring_hom.one_def NonUnitalRingHom.one_def @[simp] theorem coe_one : ⇑(1 : α →ₙ+* α) = id := rfl #align non_unital_ring_hom.coe_one NonUnitalRingHom.coe_one theorem mul_def (f g : α →ₙ+* α) : f * g = f.comp g := rfl #align non_unital_ring_hom.mul_def NonUnitalRingHom.mul_def @[simp] theorem coe_mul (f g : α →ₙ+* α) : ⇑(f * g) = f ∘ g := rfl #align non_unital_ring_hom.coe_mul NonUnitalRingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →ₙ+* γ} {f : α →ₙ+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_right NonUnitalRingHom.cancel_right @[simp] theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align non_unital_ring_hom.cancel_left NonUnitalRingHom.cancel_left end NonUnitalRingHom /-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too. This extends from both `MonoidHom` and `MonoidWithZeroHom` in order to put the fields in a sensible order, even though `MonoidWithZeroHom` already extends `MonoidHom`. -/ structure RingHom (α : Type*) (β : Type*) [NonAssocSemiring α] [NonAssocSemiring β] extends α →* β, α →+ β, α →ₙ+* β, α →*₀ β #align ring_hom RingHom /-- `α →+* β` denotes the type of ring homomorphisms from `α` to `β`. -/ infixr:25 " →+* " => RingHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid with zero homomorphism `α →*₀ β`. The `simp`-normal form is `(f : α →*₀ β)`. -/ add_decl_doc RingHom.toMonoidWithZeroHom #align ring_hom.to_monoid_with_zero_hom RingHom.toMonoidWithZeroHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a monoid homomorphism `α →* β`. The `simp`-normal form is `(f : α →* β)`. -/ add_decl_doc RingHom.toMonoidHom #align ring_hom.to_monoid_hom RingHom.toMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as an additive monoid homomorphism `α →+ β`. The `simp`-normal form is `(f : α →+ β)`. -/ add_decl_doc RingHom.toAddMonoidHom #align ring_hom.to_add_monoid_hom RingHom.toAddMonoidHom /-- Reinterpret a ring homomorphism `f : α →+* β` as a non-unital ring homomorphism `α →ₙ+* β`. The `simp`-normal form is `(f : α →ₙ+* β)`. -/ add_decl_doc RingHom.toNonUnitalRingHom #align ring_hom.to_non_unital_ring_hom RingHom.toNonUnitalRingHom section RingHomClass /-- `RingHomClass F α β` states that `F` is a type of (semi)ring homomorphisms. You should extend this class when you extend `RingHom`. This extends from both `MonoidHomClass` and `MonoidWithZeroHomClass` in order to put the fields in a sensible order, even though `MonoidWithZeroHomClass` already extends `MonoidHomClass`. -/ class RingHomClass (F : Type*) (α β : outParam (Type*)) [NonAssocSemiring α] [NonAssocSemiring β] extends MonoidHomClass F α β, AddMonoidHomClass F α β, MonoidWithZeroHomClass F α β #align ring_hom_class RingHomClass set_option linter.deprecated false in /-- Ring homomorphisms preserve `bit1`. -/ @[simp] lemma map_bit1 [NonAssocSemiring α] [NonAssocSemiring β] [RingHomClass F α β] (f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) := by simp [bit1] #align map_bit1 map_bit1 -- Porting note: marked `{}` rather than `[]` to prevent dangerous instances variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} [RingHomClass F α β] /-- Turn an element of a type `F` satisfying `RingHomClass F α β` into an actual `RingHom`. This is declared as the default coercion from `F` to `α →+* β`. -/ @[coe] def RingHomClass.toRingHom (f : F) : α →+* β := { (f : α →* β), (f : α →+ β) with } /-- Any type satisfying `RingHomClass` can be cast into `RingHom` via `RingHomClass.toRingHom`. -/ instance : CoeTC F (α →+* β) := ⟨RingHomClass.toRingHom⟩ instance (priority := 100) RingHomClass.toNonUnitalRingHomClass : NonUnitalRingHomClass F α β := { ‹RingHomClass F α β› with } #align ring_hom_class.to_non_unital_ring_hom_class RingHomClass.toNonUnitalRingHomClass end RingHomClass namespace RingHom section coe /-! Throughout this section, some `Semiring` arguments are specified with `{}` instead of `[]`. See note [implicit instance arguments]. -/ variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} instance instRingHomClass : RingHomClass (α →+* β) α β where coe f := f.toFun coe_injective' f g h := by cases f cases g congr apply FunLike.coe_injective' exact h map_add := RingHom.map_add' map_zero := RingHom.map_zero' map_mul f := f.map_mul' map_one f := f.map_one' -- Porting note: -- These helper instances are unhelpful in Lean 4, so omitting: -- /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` -- directly. -- -/ -- instance : CoeFun (α →+* β) fun _ => α → β := -- ⟨RingHom.toFun⟩ initialize_simps_projections RingHom (toFun → apply) -- Porting note: is this lemma still needed in Lean4? -- Porting note: because `f.toFun` really means `f.toMonoidHom.toOneHom.toFun` and -- `toMonoidHom_eq_coe` wants to simplify `f.toMonoidHom` to `(↑f : M →* N)`, this can't -- be a simp lemma anymore -- @[simp] theorem toFun_eq_coe (f : α →+* β) : f.toFun = f := rfl #align ring_hom.to_fun_eq_coe RingHom.toFun_eq_coe @[simp] theorem coe_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α → β) = f := rfl #align ring_hom.coe_mk RingHom.coe_mk @[simp] theorem coe_coe {F : Type*} [RingHomClass F α β] (f : F) : ((f : α →+* β) : α → β) = f := rfl #align ring_hom.coe_coe RingHom.coe_coe attribute [coe] RingHom.toMonoidHom instance coeToMonoidHom : Coe (α →+* β) (α →* β) := ⟨RingHom.toMonoidHom⟩ #align ring_hom.has_coe_monoid_hom RingHom.coeToMonoidHom -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_monoid_hom @[simp] theorem toMonoidHom_eq_coe (f : α →+* β) : f.toMonoidHom = f := rfl #align ring_hom.to_monoid_hom_eq_coe RingHom.toMonoidHom_eq_coe -- Porting note: this can't be a simp lemma anymore -- @[simp] theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f := by rfl #align ring_hom.to_monoid_with_zero_hom_eq_coe RingHom.toMonoidWithZeroHom_eq_coe @[simp] theorem coe_monoidHom_mk (f : α →* β) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : α →+* β) : α →* β) = f := rfl #align ring_hom.coe_monoid_hom_mk RingHom.coe_monoidHom_mk -- Porting note: `dsimp only` can prove this #noalign ring_hom.coe_add_monoid_hom @[simp] theorem toAddMonoidHom_eq_coe (f : α →+* β) : f.toAddMonoidHom = f := rfl #align ring_hom.to_add_monoid_hom_eq_coe RingHom.toAddMonoidHom_eq_coe @[simp] theorem coe_addMonoidHom_mk (f : α → β) (h₁ h₂ h₃ h₄) : ((⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨⟨f, h₃⟩, h₄⟩ := rfl #align ring_hom.coe_add_monoid_hom_mk RingHom.coe_addMonoidHom_mk /-- Copy of a `RingHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ def copy (f : α →+* β) (f' : α → β) (h : f' = f) : α →+* β := { f.toMonoidWithZeroHom.copy f' h, f.toAddMonoidHom.copy f' h with } #align ring_hom.copy RingHom.copy @[simp] theorem coe_copy (f : α →+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl #align ring_hom.coe_copy RingHom.coe_copy theorem copy_eq (f : α →+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f := FunLike.ext' h #align ring_hom.copy_eq RingHom.copy_eq end coe section variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} (f : α →+* β) {x y : α} theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x := FunLike.congr_fun h x #align ring_hom.congr_fun RingHom.congr_fun theorem congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y := FunLike.congr_arg f h #align ring_hom.congr_arg RingHom.congr_arg theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g := FunLike.coe_injective h #align ring_hom.coe_inj RingHom.coe_inj @[ext] theorem ext ⦃f g : α →+* β⦄ : (∀ x, f x = g x) → f = g := FunLike.ext _ _ #align ring_hom.ext RingHom.ext theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align ring_hom.ext_iff RingHom.ext_iff @[simp] theorem mk_coe (f : α →+* β) (h₁ h₂ h₃ h₄) : RingHom.mk ⟨⟨f, h₁⟩, h₂⟩ h₃ h₄ = f := ext fun _ => rfl #align ring_hom.mk_coe RingHom.mk_coe theorem coe_addMonoidHom_injective : Injective (fun f : α →+* β => (f : α →+ β)) := fun _ _ h => ext <| FunLike.congr_fun (F := α →+ β) h #align ring_hom.coe_add_monoid_hom_injective RingHom.coe_addMonoidHom_injective set_option linter.deprecated false in theorem coe_monoidHom_injective : Injective (fun f : α →+* β => (f : α →* β)) := fun _ _ h => ext <| MonoidHom.congr_fun h #align ring_hom.coe_monoid_hom_injective RingHom.coe_monoidHom_injective /-- Ring homomorphisms map zero to zero. -/ protected theorem map_zero (f : α →+* β) : f 0 = 0 := map_zero f #align ring_hom.map_zero RingHom.map_zero /-- Ring homomorphisms map one to one. -/ protected theorem map_one (f : α →+* β) : f 1 = 1 := map_one f #align ring_hom.map_one RingHom.map_one /-- Ring homomorphisms preserve addition. -/ protected theorem map_add (f : α →+* β) : ∀ a b, f (a + b) = f a + f b := map_add f #align ring_hom.map_add RingHom.map_add /-- Ring homomorphisms preserve multiplication. -/ protected theorem map_mul (f : α →+* β) : ∀ a b, f (a * b) = f a * f b := map_mul f #align ring_hom.map_mul RingHom.map_mul @[simp] theorem map_ite_zero_one {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 0 1) = ite p 0 1 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_zero_one RingHom.map_ite_zero_one @[simp] theorem map_ite_one_zero {F : Type*} [RingHomClass F α β] (f : F) (p : Prop) [Decidable p] : f (ite p 1 0) = ite p 1 0 := by split_ifs with h <;> simp [h] #align ring_hom.map_ite_one_zero RingHom.map_ite_one_zero /-- `f : α →+* β` has a trivial codomain iff `f 1 = 0`. -/ theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 := by rw [map_one, eq_comm] #align ring_hom.codomain_trivial_iff_map_one_eq_zero RingHom.codomain_trivial_iff_map_one_eq_zero /-- `f : α →+* β` has a trivial codomain iff it has a trivial range. -/ theorem codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 := f.codomain_trivial_iff_map_one_eq_zero.trans ⟨fun h x => by rw [← mul_one x, map_mul, h, mul_zero], fun h => h 1⟩ #align ring_hom.codomain_trivial_iff_range_trivial RingHom.codomain_trivial_iff_range_trivial /-- `f : α →+* β` doesn't map `1` to `0` if `β` is nontrivial -/ theorem map_one_ne_zero [Nontrivial β] : f 1 ≠ 0 := mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one #align ring_hom.map_one_ne_zero RingHom.map_one_ne_zero /-- If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. -/ theorem domain_nontrivial [Nontrivial β] : Nontrivial α := ⟨⟨1, 0, mt (fun h => show f 1 = 0 by rw [h, map_zero]) f.map_one_ne_zero⟩⟩ #align ring_hom.domain_nontrivial RingHom.domain_nontrivial theorem codomain_trivial (f : α →+* β) [h : Subsingleton α] : Subsingleton β := (subsingleton_or_nontrivial β).resolve_right fun _ => not_nontrivial_iff_subsingleton.mpr h f.domain_nontrivial #align ring_hom.codomain_trivial RingHom.codomain_trivial end /-- Ring homomorphisms preserve additive inverse. -/ protected theorem map_neg [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x : α) : f (-x) = -f x := map_neg f x #align ring_hom.map_neg RingHom.map_neg /-- Ring homomorphisms preserve subtraction. -/ protected theorem map_sub [NonAssocRing α] [NonAssocRing β] (f : α →+* β) (x y : α) : f (x - y) = f x - f y := map_sub f x y #align ring_hom.map_sub RingHom.map_sub /-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/ def mk' [NonAssocSemiring α] [NonAssocRing β] (f : α →* β) (map_add : ∀ a b, f (a + b) = f a + f b) : α →+* β := { AddMonoidHom.mk' f map_add, f with } #align ring_hom.mk' RingHom.mk' variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} /-- The identity ring homomorphism from a semiring to itself. -/ def id (α : Type*) [NonAssocSemiring α] : α →+* α := by refine' { toFun := _root_.id.. } <;> intros <;> rfl #align ring_hom.id RingHom.id instance : Inhabited (α →+* α) := ⟨id α⟩ @[simp] theorem id_apply (x : α) : RingHom.id α x = x := rfl #align ring_hom.id_apply RingHom.id_apply @[simp] theorem coe_addMonoidHom_id : (id α : α →+ α) = AddMonoidHom.id α := rfl #align ring_hom.coe_add_monoid_hom_id RingHom.coe_addMonoidHom_id @[simp] theorem coe_monoidHom_id : (id α : α →* α) = MonoidHom.id α := rfl #align ring_hom.coe_monoid_hom_id RingHom.coe_monoidHom_id variable {_ : NonAssocSemiring γ} /-- Composition of ring homomorphisms is a ring homomorphism. -/ def comp (g : β →+* γ) (f : α →+* β) : α →+* γ := { g.toNonUnitalRingHom.comp f.toNonUnitalRingHom with toFun := g ∘ f, map_one' := by simp } #align ring_hom.comp RingHom.comp /-- Composition of semiring homomorphisms is associative. -/ theorem comp_assoc {δ} {_ : NonAssocSemiring δ} (f : α →+* β) (g : β →+* γ) (h : γ →+* δ) : (h.comp g).comp f = h.comp (g.comp f) := rfl #align ring_hom.comp_assoc RingHom.comp_assoc @[simp] theorem coe_comp (hnp : β →+* γ) (hmn : α →+* β) : (hnp.comp hmn : α → γ) = hnp ∘ hmn := rfl #align ring_hom.coe_comp RingHom.coe_comp theorem comp_apply (hnp : β →+* γ) (hmn : α →+* β) (x : α) : (hnp.comp hmn : α → γ) x = hnp (hmn x) := rfl #align ring_hom.comp_apply RingHom.comp_apply @[simp] theorem comp_id (f : α →+* β) : f.comp (id α) = f := ext fun _ => rfl #align ring_hom.comp_id RingHom.comp_id @[simp] theorem id_comp (f : α →+* β) : (id β).comp f = f := ext fun _ => rfl #align ring_hom.id_comp RingHom.id_comp instance : Monoid (α →+* α) where one := id α mul := comp mul_one := comp_id one_mul := id_comp mul_assoc f g h := comp_assoc _ _ _ theorem one_def : (1 : α →+* α) = id α := rfl #align ring_hom.one_def RingHom.one_def theorem mul_def (f g : α →+* α) : f * g = f.comp g := rfl #align ring_hom.mul_def RingHom.mul_def @[simp] theorem coe_one : ⇑(1 : α →+* α) = _root_.id := rfl #align ring_hom.coe_one RingHom.coe_one @[simp] theorem coe_mul (f g : α →+* α) : ⇑(f * g) = f ∘ g := rfl #align ring_hom.coe_mul RingHom.coe_mul @[simp] theorem cancel_right {g₁ g₂ : β →+* γ} {f : α →+* β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => RingHom.ext <| hf.forall.2 (ext_iff.1 h), fun h => h ▸ rfl⟩ #align ring_hom.cancel_right RingHom.cancel_right @[simp] theorem cancel_left {g : β →+* γ} {f₁ f₂ : α →+* β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => RingHom.ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ #align ring_hom.cancel_left RingHom.cancel_left end RingHom namespace AddMonoidHom variable [CommRing α] [IsDomain α] [CommRing β] (f : β →+ α) -- Porting note: there's some disagreement over the naming scheme here. -- This could perhaps be `mkRingHom_of_mul_self_of_two_ne_zero`. -- See https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/naming.20conventions/near/315558410 /-- Make a ring homomorphism from an additive group homomorphism from a commutative ring to an integral domain that commutes with self multiplication, assumes that two is nonzero and `1` is sent to `1`. -/ def mkRingHomOfMulSelfOfTwoNeZero (h : ∀ x, f (x * x) = f x * f x) (h_two : (2 : α) ≠ 0) (h_one : f 1 = 1) : β →+* α := { f with map_one' := h_one, map_mul' := fun x y => by have hxy := h (x + y) rw [mul_add, add_mul, add_mul, f.map_add, f.map_add, f.map_add, f.map_add, h x, h y, add_mul, mul_add, mul_add, ← sub_eq_zero, add_comm (f x * f x + f (y * x)), ← sub_sub, ← sub_sub, ← sub_sub, mul_comm y x, mul_comm (f y) (f x)] at hxy simp only [add_assoc, add_sub_assoc, add_sub_cancel'_right] at hxy rw [sub_sub, ← two_mul, ← add_sub_assoc, ← two_mul, ← mul_sub, mul_eq_zero (M₀ := α), sub_eq_zero, or_iff_not_imp_left] at hxy exact hxy h_two } #align add_monoid_hom.mk_ring_hom_of_mul_self_of_two_ne_zero AddMonoidHom.mkRingHomOfMulSelfOfTwoNeZero @[simp] theorem coe_fn_mkRingHomOfMulSelfOfTwoNeZero (h h_two h_one) : (f.mkRingHomOfMulSelfOfTwoNeZero h h_two h_one : β → α) = f := rfl #align add_monoid_hom.coe_fn_mk_ring_hom_of_mul_self_of_two_ne_zero AddMonoidHom.coe_fn_mkRingHomOfMulSelfOfTwoNeZero -- Porting note: `simp` can prove this -- @[simp] theorem coe_addMonoidHom_mkRingHomOfMulSelfOfTwoNeZero (h h_two h_one) : (f.mkRingHomOfMulSelfOfTwoNeZero h h_two h_one : β →+ α) = f := by ext

rfl

theorem coe_addMonoidHom_mkRingHomOfMulSelfOfTwoNeZero (h h_two h_one) : (f.mkRingHomOfMulSelfOfTwoNeZero h h_two h_one : β →+ α) = f := by ext

Mathlib.Algebra.Ring.Hom.Defs.759_0.KyHvVYrIs9pW9ZQ

theorem coe_addMonoidHom_mkRingHomOfMulSelfOfTwoNeZero (h h_two h_one) : (f.mkRingHomOfMulSelfOfTwoNeZero h h_two h_one : β →+ α) = f

Mathlib_Algebra_Ring_Hom_Defs

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 ⊢ IsCyclotomicExtension {n} A B ↔ (∃ r, IsPrimitiveRoot r ↑n) ∧ ∀ (x : B), x ∈ 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]

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.102_0.xReI1DeVvechFQU

/-- 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}

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 ∅ 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

/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by

Mathlib.NumberTheory.Cyclotomic.Basic.109_0.xReI1DeVvechFQU

/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/ theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra 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 x : 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

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.114_0.xReI1DeVvechFQU

/-- 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) = ⊤

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 ∅ 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 => _⟩

/-- 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`.

Mathlib.NumberTheory.Cyclotomic.Basic.122_0.xReI1DeVvechFQU

/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ 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 : ⊥ = ⊤ s : ℕ+ hs : s ∈ ∅ ⊢ ∃ r, IsPrimitiveRoot r ↑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

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.122_0.xReI1DeVvechFQU

/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ 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 : ⊥ = ⊤ x : B hx : x ∈ ⊤ ⊢ x ∈ 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

/-- 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 => _⟩

Mathlib.NumberTheory.Cyclotomic.Basic.122_0.xReI1DeVvechFQU

/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ 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 : ⊥ = ⊤ x : B hx : x ∈ ⊥ ⊢ x ∈ 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

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.122_0.xReI1DeVvechFQU

/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/ theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) : IsCyclotomicExtension ∅ 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✝⁹ : CommRing A inst✝⁸ : CommRing B inst✝⁷ : Algebra A B inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L C : Type w inst✝³ : CommRing C inst✝² : Algebra A C inst✝¹ : Algebra B C inst✝ : 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

/- 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 => _⟩

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.134_0.xReI1DeVvechFQU

/-- 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

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✝⁹ : CommRing A inst✝⁸ : CommRing B inst✝⁷ : Algebra A B inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L C : Type w inst✝³ : CommRing C inst✝² : Algebra A C inst✝¹ : Algebra B C inst✝ : IsScalarTower A B C hS : IsCyclotomicExtension S A B hT : IsCyclotomicExtension T B C h : Function.Injective ⇑(algebraMap B C) n✝ : ℕ+ hn : n✝ ∈ S ∪ T ⊢ ∃ 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

/-- 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 => _⟩ ·

Mathlib.NumberTheory.Cyclotomic.Basic.134_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_1.inl n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁹ : CommRing A inst✝⁸ : CommRing B inst✝⁷ : Algebra A B inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L C : Type w inst✝³ : CommRing C inst✝² : Algebra A C inst✝¹ : Algebra B C inst✝ : IsScalarTower A B C hS : IsCyclotomicExtension S A B hT : IsCyclotomicExtension T B C h : Function.Injective ⇑(algebraMap B C) n✝ : ℕ+ hn : n✝ ∈ S ⊢ ∃ 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

/-- 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 ·

Mathlib.NumberTheory.Cyclotomic.Basic.134_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_1.inl.intro n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁹ : CommRing A inst✝⁸ : CommRing B inst✝⁷ : Algebra A B inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L C : Type w inst✝³ : CommRing C inst✝² : Algebra A C inst✝¹ : Algebra B C inst✝ : IsScalarTower A B C hS : IsCyclotomicExtension S A B hT : IsCyclotomicExtension T B C h : Function.Injective ⇑(algebraMap B C) n✝ : ℕ+ hn : n✝ ∈ S b : B hb : IsPrimitiveRoot b ↑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, _⟩

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.134_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_1.inl.intro n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁹ : CommRing A inst✝⁸ : CommRing B inst✝⁷ : Algebra A B inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L C : Type w inst✝³ : CommRing C inst✝² : Algebra A C inst✝¹ : Algebra B C inst✝ : IsScalarTower A B C hS : IsCyclotomicExtension S A B hT : IsCyclotomicExtension T B C h : Function.Injective ⇑(algebraMap B C) n✝ : ℕ+ hn : n✝ ∈ S b : B hb : IsPrimitiveRoot b ↑n✝ ⊢ IsPrimitiveRoot ((algebraMap B C) b) ↑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

/-- 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, _⟩

Mathlib.NumberTheory.Cyclotomic.Basic.134_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_1.inr n : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁹ : CommRing A inst✝⁸ : CommRing B inst✝⁷ : Algebra A B inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L C : Type w inst✝³ : CommRing C inst✝² : Algebra A C inst✝¹ : Algebra B C inst✝ : IsScalarTower A B C hS : IsCyclotomicExtension S A B hT : IsCyclotomicExtension T B C h : Function.Injective ⇑(algebraMap B C) n✝ : ℕ+ hn : n✝ ∈ T ⊢ ∃ 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

/-- 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 ·

Mathlib.NumberTheory.Cyclotomic.Basic.134_0.xReI1DeVvechFQU

/-- 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

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✝⁹ : CommRing A inst✝⁸ : CommRing B inst✝⁷ : Algebra A B inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L C : Type w inst✝³ : CommRing C inst✝² : Algebra A C inst✝¹ : Algebra B C inst✝ : IsScalarTower A B C hS : IsCyclotomicExtension S A B hT : IsCyclotomicExtension T B C h : Function.Injective ⇑(algebraMap B C) x : C ⊢ x ∈ adjoin A {b | ∃ n ∈ S ∪ T, 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

/-- 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 ·

Mathlib.NumberTheory.Cyclotomic.Basic.134_0.xReI1DeVvechFQU

/-- 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

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✝⁹ : CommRing A inst✝⁸ : CommRing B inst✝⁷ : Algebra A B inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L C : Type w inst✝³ : CommRing C inst✝² : Algebra A C inst✝¹ : Algebra B C inst✝ : IsScalarTower A B C hS : IsCyclotomicExtension S A B hT : IsCyclotomicExtension T B C h : Function.Injective ⇑(algebraMap B C) x : C b : B ⊢ (algebraMap B C) b ∈ adjoin A {b | ∃ n ∈ S ∪ T, 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

/-- 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 ·

Mathlib.NumberTheory.Cyclotomic.Basic.134_0.xReI1DeVvechFQU

/-- 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

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✝⁹ : CommRing A inst✝⁸ : CommRing B inst✝⁷ : Algebra A B inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L C : Type w inst✝³ : CommRing C inst✝² : Algebra A C inst✝¹ : Algebra B C inst✝ : IsScalarTower A B C hS : IsCyclotomicExtension S A B hT : IsCyclotomicExtension T B C h : Function.Injective ⇑(algebraMap B C) x : C b : B f : B →ₐ[A] C := IsScalarTower.toAlgHom A B C ⊢ (algebraMap B C) b ∈ adjoin A {b | ∃ n ∈ S ∪ T, 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⟩

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.134_0.xReI1DeVvechFQU

/-- 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

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✝⁹ : CommRing A inst✝⁸ : CommRing B inst✝⁷ : Algebra A B inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L C : Type w inst✝³ : CommRing C inst✝² : Algebra A C inst✝¹ : Algebra B C inst✝ : IsScalarTower A B C hS : IsCyclotomicExtension S A B hT : IsCyclotomicExtension T B C h : Function.Injective ⇑(algebraMap B C) x : C b : B f : B →ₐ[A] C := IsScalarTower.toAlgHom A B C hb : f b ∈ Subalgebra.map f (adjoin A {b | ∃ a ∈ S, b ^ ↑a = 1}) ⊢ (algebraMap B C) b ∈ adjoin A {b | ∃ n ∈ S ∪ T, 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

/-- 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⟩

Mathlib.NumberTheory.Cyclotomic.Basic.134_0.xReI1DeVvechFQU

/-- 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

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✝⁹ : CommRing A inst✝⁸ : CommRing B inst✝⁷ : Algebra A B inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L C : Type w inst✝³ : CommRing C inst✝² : Algebra A C inst✝¹ : Algebra B C inst✝ : IsScalarTower A B C hS : IsCyclotomicExtension S A B hT : IsCyclotomicExtension T B C h : Function.Injective ⇑(algebraMap B C) x : C b : B f : B →ₐ[A] C := IsScalarTower.toAlgHom A B C hb : (algebraMap B C) b ∈ adjoin A (⇑f '' {b | ∃ a ∈ S, b ^ ↑a = 1}) ⊢ (algebraMap B C) b ∈ adjoin A {b | ∃ n ∈ S ∪ T, 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

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.134_0.xReI1DeVvechFQU

/-- 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

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✝⁹ : CommRing A inst✝⁸ : CommRing B inst✝⁷ : Algebra A B inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L C : Type w inst✝³ : CommRing C inst✝² : Algebra A C inst✝¹ : Algebra B C inst✝ : IsScalarTower A B C hS : IsCyclotomicExtension S A B hT : IsCyclotomicExtension T B C h : Function.Injective ⇑(algebraMap B C) x : C b : B f : B →ₐ[A] C := IsScalarTower.toAlgHom A B C hb : (algebraMap B C) b ∈ adjoin A (⇑f '' {b | ∃ a ∈ S, b ^ ↑a = 1}) y : C hy : y ∈ ⇑f '' {b | ∃ a ∈ S, b ^ ↑a = 1} ⊢ y ∈ {b | ∃ n ∈ S ∪ T, 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

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.134_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_2.intro.intro.intro n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁹ : CommRing A inst✝⁸ : CommRing B inst✝⁷ : Algebra A B inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L C : Type w inst✝³ : CommRing C inst✝² : Algebra A C inst✝¹ : Algebra B C inst✝ : IsScalarTower A B C hS : IsCyclotomicExtension S A B hT : IsCyclotomicExtension T B C h : Function.Injective ⇑(algebraMap B C) x : C b : B f : B →ₐ[A] C := IsScalarTower.toAlgHom A B C hb : (algebraMap B C) b ∈ adjoin A (⇑f '' {b | ∃ a ∈ S, b ^ ↑a = 1}) y : C b₁ : B h₁ : f b₁ = y n : ℕ+ hn : n ∈ S ∧ b₁ ^ ↑n = 1 ⊢ y ∈ {b | ∃ n ∈ S ∪ T, 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]⟩⟩

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.134_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

n✝ : ℕ+ S T : Set ℕ+ A : Type u B : Type v K : Type w L : Type z inst✝⁹ : CommRing A inst✝⁸ : CommRing B inst✝⁷ : Algebra A B inst✝⁶ : Field K inst✝⁵ : Field L inst✝⁴ : Algebra K L C : Type w inst✝³ : CommRing C inst✝² : Algebra A C inst✝¹ : Algebra B C inst✝ : IsScalarTower A B C hS : IsCyclotomicExtension S A B hT : IsCyclotomicExtension T B C h : Function.Injective ⇑(algebraMap B C) x : C b : B f : B →ₐ[A] C := IsScalarTower.toAlgHom A B C hb : (algebraMap B C) b ∈ adjoin A (⇑f '' {b | ∃ a ∈ S, b ^ ↑a = 1}) y : C b₁ : B h₁ : f b₁ = y n : ℕ+ hn : n ∈ S ∧ b₁ ^ ↑n = 1 ⊢ y ^ ↑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]

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.134_0.xReI1DeVvechFQU

/-- 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

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✝ : Subsingleton B ⊢ IsCyclotomicExtension S A B ↔ S = ∅ ∨ S = {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

@[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by

Mathlib.NumberTheory.Cyclotomic.Basic.156_0.xReI1DeVvechFQU

@[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1}

Mathlib_NumberTheory_Cyclotomic_Basic

case mp 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✝ : Subsingleton B ⊢ IsCyclotomicExtension S A B → S = ∅ ∨ S = {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, -⟩

@[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor ·

Mathlib.NumberTheory.Cyclotomic.Basic.156_0.xReI1DeVvechFQU

@[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1}

Mathlib_NumberTheory_Cyclotomic_Basic

case mp.mk 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✝ : Subsingleton B hprim : ∀ {n : ℕ+}, n ∈ S → ∃ r, IsPrimitiveRoot r ↑n ⊢ S = ∅ ∨ S = {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]

@[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩

Mathlib.NumberTheory.Cyclotomic.Basic.156_0.xReI1DeVvechFQU

@[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1}

Mathlib_NumberTheory_Cyclotomic_Basic

case mp.mk 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✝ : Subsingleton B hprim : ∀ {n : ℕ+}, n ∈ S → ∃ r, IsPrimitiveRoot r ↑n ⊢ S ⊆ {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

@[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by constructor · rintro ⟨hprim, -⟩ rw [← subset_singleton_iff_eq]

Mathlib.NumberTheory.Cyclotomic.Basic.156_0.xReI1DeVvechFQU

@[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1}

Mathlib_NumberTheory_Cyclotomic_Basic

case mp.mk 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✝ : Subsingleton B hprim : ∀ {n : ℕ+}, n ∈ S → ∃ r, IsPrimitiveRoot r ↑n t : ℕ+ ht : t ∈ S ⊢ t ∈ {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

@[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

Mathlib.NumberTheory.Cyclotomic.Basic.156_0.xReI1DeVvechFQU

@[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1}

Mathlib_NumberTheory_Cyclotomic_Basic

case mp.mk.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✝ : Subsingleton B hprim : ∀ {n : ℕ+}, n ∈ S → ∃ r, IsPrimitiveRoot r ↑n t : ℕ+ ht : t ∈ S ζ : B hζ : IsPrimitiveRoot ζ ↑t ⊢ t ∈ {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]

@[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

Mathlib.NumberTheory.Cyclotomic.Basic.156_0.xReI1DeVvechFQU

@[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1}

Mathlib_NumberTheory_Cyclotomic_Basic

case mp.mk.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✝ : Subsingleton B hprim : ∀ {n : ℕ+}, n ∈ S → ∃ r, IsPrimitiveRoot r ↑n t : ℕ+ ht : t ∈ S ζ : B hζ : IsPrimitiveRoot ζ ↑t ⊢ ↑t = 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)

@[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]

Mathlib.NumberTheory.Cyclotomic.Basic.156_0.xReI1DeVvechFQU

@[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1}

Mathlib_NumberTheory_Cyclotomic_Basic

case mpr 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✝ : Subsingleton B ⊢ S = ∅ ∨ S = {1} → IsCyclotomicExtension S 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)

@[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) ·

Mathlib.NumberTheory.Cyclotomic.Basic.156_0.xReI1DeVvechFQU

@[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1}

Mathlib_NumberTheory_Cyclotomic_Basic

case mpr.inl n : ℕ+ 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✝ : Subsingleton 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 ∈ ⊤)⟩

@[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. ·

Mathlib.NumberTheory.Cyclotomic.Basic.156_0.xReI1DeVvechFQU

@[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1}

Mathlib_NumberTheory_Cyclotomic_Basic

n : ℕ+ 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✝ : Subsingleton B x : B ⊢ x ∈ 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 ∈ ⊤)

@[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

Mathlib.NumberTheory.Cyclotomic.Basic.156_0.xReI1DeVvechFQU

@[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1}

Mathlib_NumberTheory_Cyclotomic_Basic

case mpr.inr n : ℕ+ 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✝ : Subsingleton 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]

@[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 ∈ ⊤)⟩ ·

Mathlib.NumberTheory.Cyclotomic.Basic.156_0.xReI1DeVvechFQU

@[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1}

Mathlib_NumberTheory_Cyclotomic_Basic

case mpr.inr n : ℕ+ 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✝ : Subsingleton B ⊢ (∃ r, IsPrimitiveRoot r ↑1) ∧ ∀ (x : B), x ∈ adjoin A {b | b ^ ↑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 ∈ ⊤)⟩

@[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]

Mathlib.NumberTheory.Cyclotomic.Basic.156_0.xReI1DeVvechFQU

@[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1}

Mathlib_NumberTheory_Cyclotomic_Basic

n : ℕ+ 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✝ : Subsingleton B x : B ⊢ x ∈ adjoin A {b | b ^ ↑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 ∈ ⊤)

@[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

Mathlib.NumberTheory.Cyclotomic.Basic.156_0.xReI1DeVvechFQU

@[nontriviality] theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {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 ∪ T) A B ⊢ IsCyclotomicExtension T (↥(adjoin A {b | ∃ a ∈ S, b ^ ↑a = 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⟩

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.173_0.xReI1DeVvechFQU

/-- 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

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 ∪ T) A B ⊢ {b | ∃ n ∈ S ∪ T, b ^ ↑n = 1} = {b | ∃ n ∈ S, b ^ ↑n = 1} ∪ {b | ∃ n ∈ T, 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 _ _

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.173_0.xReI1DeVvechFQU

/-- 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

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 h : IsCyclotomicExtension (S ∪ T) A B ⊢ {b | ∃ n ∈ S ∪ T, b ^ ↑n = 1} ≤ {b | ∃ n ∈ S, b ^ ↑n = 1} ∪ {b | ∃ n ∈ T, 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⟩

/-- 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 _ _ ·

Mathlib.NumberTheory.Cyclotomic.Basic.173_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_1.intro.intro.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 h : IsCyclotomicExtension (S ∪ T) A B x : B n : ℕ+ hnpow : x ^ ↑n = 1 hn₁ : n ∈ S ⊢ x ∈ {b | ∃ n ∈ S, b ^ ↑n = 1} ∪ {b | ∃ n ∈ T, 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

/-- 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⟩ ·

Mathlib.NumberTheory.Cyclotomic.Basic.173_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_1.intro.intro.inl.h 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 ∪ T) A B x : B n : ℕ+ hnpow : x ^ ↑n = 1 hn₁ : n ∈ S ⊢ x ∈ {b | ∃ n ∈ S, 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⟩

/-- 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;

Mathlib.NumberTheory.Cyclotomic.Basic.173_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_1.intro.intro.inr 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 ∪ T) A B x : B n : ℕ+ hnpow : x ^ ↑n = 1 hn₂ : n ∈ T ⊢ x ∈ {b | ∃ n ∈ S, b ^ ↑n = 1} ∪ {b | ∃ n ∈ T, 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

/-- 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⟩ ·

Mathlib.NumberTheory.Cyclotomic.Basic.173_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_1.intro.intro.inr.h 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 ∪ T) A B x : B n : ℕ+ hnpow : x ^ ↑n = 1 hn₂ : n ∈ T ⊢ x ∈ {b | ∃ n ∈ T, 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⟩

/-- 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;

Mathlib.NumberTheory.Cyclotomic.Basic.173_0.xReI1DeVvechFQU

/-- 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

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 h : IsCyclotomicExtension (S ∪ T) A B ⊢ {b | ∃ n ∈ S, b ^ ↑n = 1} ∪ {b | ∃ n ∈ T, b ^ ↑n = 1} ≤ {b | ∃ n ∈ S ∪ T, 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⟩)

/-- 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⟩ ·

Mathlib.NumberTheory.Cyclotomic.Basic.173_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_2.inl.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 ∪ T) A B x : B n : ℕ+ hn : n ∈ S ∧ x ^ ↑n = 1 ⊢ x ∈ {b | ∃ n ∈ S ∪ T, 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⟩

/-- 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⟩) ·

Mathlib.NumberTheory.Cyclotomic.Basic.173_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_2.inr.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 ∪ T) A B x : B n : ℕ+ hn : n ∈ T ∧ x ^ ↑n = 1 ⊢ x ∈ {b | ∃ n ∈ S ∪ T, 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⟩

/-- 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⟩ ·

Mathlib.NumberTheory.Cyclotomic.Basic.173_0.xReI1DeVvechFQU

/-- 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

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 ∪ T) A B this : {b | ∃ n ∈ S ∪ T, b ^ ↑n = 1} = {b | ∃ n ∈ S, b ^ ↑n = 1} ∪ {b | ∃ n ∈ T, b ^ ↑n = 1} ⊢ IsCyclotomicExtension T (↥(adjoin A {b | ∃ a ∈ S, b ^ ↑a = 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 => _⟩

/-- 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⟩

Mathlib.NumberTheory.Cyclotomic.Basic.173_0.xReI1DeVvechFQU

/-- 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

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 ∪ T) A B this : {b | ∃ n ∈ S ∪ T, b ^ ↑n = 1} = {b | ∃ n ∈ S, b ^ ↑n = 1} ∪ {b | ∃ n ∈ T, b ^ ↑n = 1} b : B ⊢ b ∈ adjoin ↥(adjoin A {b | ∃ a ∈ S, b ^ ↑a = 1}) {b | ∃ n ∈ T, 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

/-- 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 => _⟩

Mathlib.NumberTheory.Cyclotomic.Basic.173_0.xReI1DeVvechFQU

/-- 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

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 this : {b | ∃ n ∈ S ∪ T, b ^ ↑n = 1} = {b | ∃ n ∈ S, b ^ ↑n = 1} ∪ {b | ∃ n ∈ T, b ^ ↑n = 1} b : B h : b ∈ adjoin A {b | ∃ n ∈ S ∪ T, b ^ ↑n = 1} ⊢ b ∈ adjoin ↥(adjoin A {b | ∃ a ∈ S, b ^ ↑a = 1}) {b | ∃ n ∈ T, 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

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.173_0.xReI1DeVvechFQU

/-- 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

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 T A B hS : S ⊆ T ⊢ IsCyclotomicExtension S A ↥(adjoin A {b | ∃ a ∈ S, 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 => _⟩

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.193_0.xReI1DeVvechFQU

/-- 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})

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 h : IsCyclotomicExtension T A B hS : S ⊆ T n : ℕ+ hn : n ∈ S ⊢ ∃ 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)

/-- 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 => _⟩ ·

Mathlib.NumberTheory.Cyclotomic.Basic.193_0.xReI1DeVvechFQU

/-- 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})

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_1.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 T A B hS : S ⊆ T n : ℕ+ hn : n ∈ S b : B hb : IsPrimitiveRoot b ↑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⟩⟩, _⟩

/-- 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)

Mathlib.NumberTheory.Cyclotomic.Basic.193_0.xReI1DeVvechFQU

/-- 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})

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_1.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 T A B hS : S ⊆ T n : ℕ+ hn : n ∈ S b : B hb : IsPrimitiveRoot b ↑n ⊢ IsPrimitiveRoot { val := b, property := (_ : b ∈ ↑(adjoin A {b | ∃ a ∈ S, b ^ ↑a = 1})) } ↑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]

/-- 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⟩⟩, _⟩

Mathlib.NumberTheory.Cyclotomic.Basic.193_0.xReI1DeVvechFQU

/-- 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})

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 h : IsCyclotomicExtension T A B hS : S ⊆ T b : ↥(adjoin A {b | ∃ a ∈ S, b ^ ↑a = 1}) ⊢ b ∈ adjoin A {b | ∃ n ∈ S, 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)

/-- 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] ·

Mathlib.NumberTheory.Cyclotomic.Basic.193_0.xReI1DeVvechFQU

/-- 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})

Mathlib_NumberTheory_Cyclotomic_Basic

case h.e'_5 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 T A B hS : S ⊆ T b : ↥(adjoin A {b | ∃ a ∈ S, b ^ ↑a = 1}) ⊢ adjoin A {b | ∃ n ∈ S, 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]

/-- 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)

Mathlib.NumberTheory.Cyclotomic.Basic.193_0.xReI1DeVvechFQU

/-- 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})

Mathlib_NumberTheory_Cyclotomic_Basic

case h.e'_5 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 T A B hS : S ⊆ T b : ↥(adjoin A {b | ∃ a ∈ S, b ^ ↑a = 1}) ⊢ adjoin A {b | ∃ n ∈ S, b ^ ↑n = 1} = adjoin A {a | ∃ a_1 ∈ S, ↑a ^ ↑a_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

/-- 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]

Mathlib.NumberTheory.Cyclotomic.Basic.193_0.xReI1DeVvechFQU

/-- 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})

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 : ∀ s ∈ S, n ∣ s hS : Set.Nonempty S H : IsCyclotomicExtension S A B ⊢ IsCyclotomicExtension (S ∪ {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 => _, _⟩

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.209_0.xReI1DeVvechFQU

/-- 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

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 h : ∀ s ∈ S, n ∣ s hS : Set.Nonempty S H : IsCyclotomicExtension S A B s : ℕ+ hs : s ∈ S ∪ {n} ⊢ ∃ r, IsPrimitiveRoot r ↑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

/-- 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 => _, _⟩ ·

Mathlib.NumberTheory.Cyclotomic.Basic.209_0.xReI1DeVvechFQU

/-- 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

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 h : ∀ s ∈ S, n ∣ s hS : Set.Nonempty S H : IsCyclotomicExtension S A B s : ℕ+ hs : s ∈ S ∨ s = n ⊢ ∃ r, IsPrimitiveRoot r ↑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

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.209_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_1.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 h : ∀ s ∈ S, n ∣ s hS : Set.Nonempty S H : IsCyclotomicExtension S A B s : ℕ+ hs : s ∈ S ⊢ ∃ r, IsPrimitiveRoot r ↑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

/-- 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 ·

Mathlib.NumberTheory.Cyclotomic.Basic.209_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_1.inr 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 H : IsCyclotomicExtension S A B s : ℕ+ h : ∀ s_1 ∈ S, s ∣ s_1 ⊢ ∃ r, IsPrimitiveRoot r ↑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

/-- 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 ·

Mathlib.NumberTheory.Cyclotomic.Basic.209_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_1.inr.intro 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 A B s : ℕ+ h : ∀ s_1 ∈ S, s ∣ s_1 m : ℕ+ hm : m ∈ S ⊢ ∃ r, IsPrimitiveRoot r ↑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

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.209_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_1.inr.intro.intro 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 A B s : ℕ+ h : ∀ s_1 ∈ S, s ∣ s_1 x : ℕ+ hm : s * x ∈ S ⊢ ∃ r, IsPrimitiveRoot r ↑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

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.209_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_1.inr.intro.intro.intro 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 A B s : ℕ+ h : ∀ s_1 ∈ S, s ∣ s_1 x : ℕ+ hm : s * x ∈ S ζ : B hζ : IsPrimitiveRoot ζ ↑(s * x) ⊢ ∃ r, IsPrimitiveRoot r ↑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), _⟩

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.209_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_1.inr.intro.intro.intro 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 A B s : ℕ+ h : ∀ s_1 ∈ S, s ∣ s_1 x : ℕ+ hm : s * x ∈ S ζ : B hζ : IsPrimitiveRoot ζ ↑(s * x) ⊢ IsPrimitiveRoot (ζ ^ ↑x) ↑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)

/-- 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), _⟩

Mathlib.NumberTheory.Cyclotomic.Basic.209_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case h.e'_4 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 A B s : ℕ+ h : ∀ s_1 ∈ S, s ∣ s_1 x : ℕ+ hm : s * x ∈ S ζ : B hζ : IsPrimitiveRoot ζ ↑(s * x) ⊢ ↑s = ↑(s * x) / ↑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]

/-- 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)

Mathlib.NumberTheory.Cyclotomic.Basic.209_0.xReI1DeVvechFQU

/-- 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

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 h : ∀ s ∈ S, n ∣ s hS : Set.Nonempty S H : IsCyclotomicExtension S A B ⊢ adjoin A {b | ∃ n_1 ∈ S ∪ {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 _

/-- 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] ·

Mathlib.NumberTheory.Cyclotomic.Basic.209_0.xReI1DeVvechFQU

/-- 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

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 h : ∀ s ∈ S, n ∣ s hS : Set.Nonempty S H : IsCyclotomicExtension S A B ⊢ ⊤ ≤ adjoin A {b | ∃ n_1 ∈ S ∪ {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]

/-- 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 _

Mathlib.NumberTheory.Cyclotomic.Basic.209_0.xReI1DeVvechFQU

/-- 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

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 h : ∀ s ∈ S, n ∣ s hS : Set.Nonempty S H : IsCyclotomicExtension S A B ⊢ adjoin A {b | ∃ n ∈ S, b ^ ↑n = 1} ≤ adjoin A {b | ∃ n_1 ∈ S ∪ {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 => _

/-- 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]

Mathlib.NumberTheory.Cyclotomic.Basic.209_0.xReI1DeVvechFQU

/-- 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

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 h : ∀ s ∈ S, n ∣ s hS : Set.Nonempty S H : IsCyclotomicExtension S A B x : B hx : x ∈ {b | ∃ n ∈ S, b ^ ↑n = 1} ⊢ x ∈ {b | ∃ n_1 ∈ S ∪ {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 ⊢

/-- 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 => _

Mathlib.NumberTheory.Cyclotomic.Basic.209_0.xReI1DeVvechFQU

/-- 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

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 h : ∀ s ∈ S, n ∣ s hS : Set.Nonempty S H : IsCyclotomicExtension S A B x : B hx : ∃ n ∈ S, x ^ ↑n = 1 ⊢ ∃ n_1, (n_1 = n ∨ n_1 ∈ S) ∧ x ^ ↑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

/-- 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 ⊢

Mathlib.NumberTheory.Cyclotomic.Basic.209_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_2.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 : ∀ s ∈ S, n ∣ s hS : Set.Nonempty S H : IsCyclotomicExtension S A B x : B m : ℕ+ hm : m ∈ S ∧ x ^ ↑m = 1 ⊢ ∃ n_1, (n_1 = n ∨ n_1 ∈ S) ∧ x ^ ↑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⟩⟩

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.209_0.xReI1DeVvechFQU

/-- 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

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 : ∀ s ∈ S, n ∣ s hS : Set.Nonempty S ⊢ IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {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 => _, _⟩⟩

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.231_0.xReI1DeVvechFQU

/-- 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

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 h : ∀ s ∈ S, n ∣ s hS : Set.Nonempty S H : IsCyclotomicExtension (S ∪ {n}) A B s : ℕ+ hs : s ∈ S ⊢ ∃ r, IsPrimitiveRoot r ↑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)

/-- 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 => _, _⟩⟩ ·

Mathlib.NumberTheory.Cyclotomic.Basic.231_0.xReI1DeVvechFQU

/-- 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

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 h : ∀ s ∈ S, n ∣ s hS : Set.Nonempty S H : IsCyclotomicExtension (S ∪ {n}) A B ⊢ adjoin A {b | ∃ n ∈ S, 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]

/-- 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) ·

Mathlib.NumberTheory.Cyclotomic.Basic.231_0.xReI1DeVvechFQU

/-- 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

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 h : ∀ s ∈ S, n ∣ s hS : Set.Nonempty S H : IsCyclotomicExtension (S ∪ {n}) A B ⊢ adjoin A {b | ∃ n_1 ∈ S ∪ {n}, b ^ ↑n_1 = 1} ≤ adjoin A {b | ∃ n ∈ S, 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 => _

/-- 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]

Mathlib.NumberTheory.Cyclotomic.Basic.231_0.xReI1DeVvechFQU

/-- 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

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 h : ∀ s ∈ S, n ∣ s hS : Set.Nonempty S H : IsCyclotomicExtension (S ∪ {n}) A B x : B hx : x ∈ {b | ∃ n_1 ∈ S ∪ {n}, b ^ ↑n_1 = 1} ⊢ x ∈ {b | ∃ n ∈ S, 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 ⊢

/-- 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 => _

Mathlib.NumberTheory.Cyclotomic.Basic.231_0.xReI1DeVvechFQU

/-- 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

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 h : ∀ s ∈ S, n ∣ s hS : Set.Nonempty S H : IsCyclotomicExtension (S ∪ {n}) A B x : B hx : ∃ n_1, (n_1 = n ∨ n_1 ∈ S) ∧ x ^ ↑n_1 = 1 ⊢ ∃ n ∈ S, 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

/-- 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 ⊢

Mathlib.NumberTheory.Cyclotomic.Basic.231_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_2.intro.intro.inl 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 x : B m : ℕ+ hxpow : x ^ ↑m = 1 h : ∀ s ∈ S, m ∣ s H : IsCyclotomicExtension (S ∪ {m}) A B ⊢ ∃ n ∈ S, 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

/-- 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 ·

Mathlib.NumberTheory.Cyclotomic.Basic.231_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_2.intro.intro.inl.intro 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 x : B m : ℕ+ hxpow : x ^ ↑m = 1 h : ∀ s ∈ S, m ∣ s H : IsCyclotomicExtension (S ∪ {m}) A B y : ℕ+ hy : y ∈ S ⊢ ∃ n ∈ S, 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, _⟩⟩

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.231_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_2.intro.intro.inl.intro 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 x : B m : ℕ+ hxpow : x ^ ↑m = 1 h : ∀ s ∈ S, m ∣ s H : IsCyclotomicExtension (S ∪ {m}) A B y : ℕ+ hy : y ∈ S ⊢ x ^ ↑y = 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

/-- 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, _⟩⟩

Mathlib.NumberTheory.Cyclotomic.Basic.231_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_2.intro.intro.inl.intro.intro 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 x : B m : ℕ+ hxpow : x ^ ↑m = 1 h : ∀ s ∈ S, m ∣ s H : IsCyclotomicExtension (S ∪ {m}) A B z : ℕ+ hy : m * z ∈ S ⊢ x ^ ↑(m * z) = 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]

/-- 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

Mathlib.NumberTheory.Cyclotomic.Basic.231_0.xReI1DeVvechFQU

/-- 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

Mathlib_NumberTheory_Cyclotomic_Basic

case refine'_2.intro.intro.inr 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 : ∀ s ∈ S, n ∣ s hS : Set.Nonempty S H : IsCyclotomicExtension (S ∪ {n}) A B x : B m : ℕ+ hxpow : x ^ ↑m = 1 hm : m ∈ S ⊢ ∃ n ∈ S, 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⟩⟩

/-- 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] ·

Mathlib.NumberTheory.Cyclotomic.Basic.231_0.xReI1DeVvechFQU

/-- 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

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 ⊢ 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

/-- `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

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