state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case inl
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
hS : Set.Nonempty S
⊢ IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS | /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· | Mathlib.NumberTheory.Cyclotomic.Basic.251_0.xReI1DeVvechFQU | /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case inr
n : ℕ+
T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
⊢ IsCyclotomicExtension ∅ A B ↔ IsCyclotomicExtension (∅ ∪ {1}) A B | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rw [empty_union] | /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
| Mathlib.NumberTheory.Cyclotomic.Basic.251_0.xReI1DeVvechFQU | /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case inr
n : ℕ+
T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
⊢ IsCyclotomicExtension ∅ A B ↔ IsCyclotomicExtension {1} A B | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' ⟨fun H => _, fun H => _⟩ | /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty... | Mathlib.NumberTheory.Cyclotomic.Basic.251_0.xReI1DeVvechFQU | /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case inr.refine'_1
n : ℕ+
T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
H : IsCyclotomicExtension ∅ A B
⊢ IsCyclotomicExtension {1} A B | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩ | /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty... | Mathlib.NumberTheory.Cyclotomic.Basic.251_0.xReI1DeVvechFQU | /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
H : IsCyclotomicExtension ∅ A B
s : ℕ+
hs : s ∈ {1}
⊢ IsPrimitiveRoot 1 ↑s | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simp [mem_singleton_iff.1 hs] | /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty... | Mathlib.NumberTheory.Cyclotomic.Basic.251_0.xReI1DeVvechFQU | /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case inr.refine'_1
n : ℕ+
T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
H : IsCyclotomicExtension ∅ A B
⊢ adjoin A {b | ∃ n ∈ {1}, b ^ ↑n = 1} = ⊤ | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simp [adjoin_singleton_one, empty] | /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty... | Mathlib.NumberTheory.Cyclotomic.Basic.251_0.xReI1DeVvechFQU | /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case inr.refine'_2
n : ℕ+
T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
H : IsCyclotomicExtension {1} A B
⊢ IsCyclotomicExtension ∅ A B | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩ | /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty... | Mathlib.NumberTheory.Cyclotomic.Basic.251_0.xReI1DeVvechFQU | /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case inr.refine'_2
n : ℕ+
T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
H : IsCyclotomicExtension {1} A B
⊢ adjoin A {b | ∃ n ∈ ∅, b ^ ↑n = 1} = ⊤ | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simp [@singleton_one A B _ _ _ H] | /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty... | Mathlib.NumberTheory.Cyclotomic.Basic.251_0.xReI1DeVvechFQU | /-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
h : ⊥ = ⊤
⊢ IsCyclotomicExtension {1} A B | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h) | /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
| Mathlib.NumberTheory.Cyclotomic.Basic.266_0.xReI1DeVvechFQU | /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case h.e'_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
h : ⊥ = ⊤
⊢ {1} = ∅ ∪ {1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simp | /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
| Mathlib.NumberTheory.Cyclotomic.Basic.266_0.xReI1DeVvechFQU | /-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
C : Type u_1
inst✝¹ : CommRing C
inst✝ : Algebra A C
h : IsCyclotomicExtension S A B
f : B ≃ₐ[A] C
⊢ IsCyclotomicExtension S A C | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra | /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
| Mathlib.NumberTheory.Cyclotomic.Basic.281_0.xReI1DeVvechFQU | /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
C : Type u_1
inst✝¹ : CommRing C
inst✝ : Algebra A C
h : IsCyclotomicExtension S A B
f : B ≃ₐ[A] C
this : Algebra B C := RingHom.toAlgebra ↑... | /-
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.c... | haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective | /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
| Mathlib.NumberTheory.Cyclotomic.Basic.281_0.xReI1DeVvechFQU | /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
C : Type u_1
inst✝¹ : CommRing C
inst✝ : Algebra A C
h : IsCyclotomicExtension S A B
f : B ≃ₐ[A] C
this✝ : Algebra B C := RingHom.toAlgebra ... | /-
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.c... | haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom | /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : I... | Mathlib.NumberTheory.Cyclotomic.Basic.281_0.xReI1DeVvechFQU | /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
C : Type u_1
inst✝¹ : CommRing C
inst✝ : Algebra A C
h : IsCyclotomicExtension S A B
f : B ≃ₐ[A] C
this✝¹ : Algebra B C := RingHom.toAlgebra... | /-
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.c... | exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective) | /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : I... | Mathlib.NumberTheory.Cyclotomic.Basic.281_0.xReI1DeVvechFQU | /-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
h : IsCyclotomicExtension {n} A B
inst✝ : IsDomain B
⊢ NeZero ↑↑n | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h | protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.292_0.xReI1DeVvechFQU | protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
h : IsCyclotomicExtension {n} A B
inst✝ : IsDomain B
r : B
hr : IsPrimitiveRoot r ↑n
⊢ NeZero ↑↑n | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | exact hr.neZero' | protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
| Mathlib.NumberTheory.Cyclotomic.Basic.292_0.xReI1DeVvechFQU | protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : IsCyclotomicExtension {n} A B
inst✝ : IsDomain B
⊢ NeZero ↑↑n | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | haveI := IsCyclotomicExtension.neZero n A B | protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.298_0.xReI1DeVvechFQU | protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : IsCyclotomicExtension {n} A B
inst✝ : IsDomain B
this : NeZero ↑↑n
⊢ NeZero ↑↑n | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | exact NeZero.nat_of_neZero (algebraMap A B) | protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
| Mathlib.NumberTheory.Cyclotomic.Basic.298_0.xReI1DeVvechFQU | protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
h : IsCyclotomicExtension {n} A B
⊢ Module.Finite A B | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => b... | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
| Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
h : IsCyclotomicExtension {n} A B
⊢ Module.Finite A B | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
| Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
h : IsCyclotomicExtension {n} A B
⊢ Submodule.FG (Subalgebra.toSubmodule (adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1})) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' fg_adjoin_of_finite _ fun b hb => _ | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
| Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
h : IsCyclotomicExtension {n} A B
⊢ Set.Finite {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simp only [mem_singleton_iff, exists_eq_left] | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· | Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
h : IsCyclotomicExtension {n} A B
⊢ Set.Finite {b | b ^ ↑n = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩ | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
| Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
h✝ : IsCyclotomicExtension {n} A B
x : B
h : x ∈ {b | b ^ ↑n = 1}
⊢ x ∈ ↑(Multiset.toFinset (nthRoots (↑n) 1)) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simpa using h | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ ... | Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
h✝ : IsCyclotomicExtension {n} A B
x : B
h : x ∈ ↑(Multiset.toFinset (nthRoots (↑n) 1))
⊢ x ∈ {b | b ^ ↑n = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simpa using h | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ ... | Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
h : IsCyclotomicExtension {n} A B
this : {b | b ^ ↑n = 1} = ↑(Multiset.toFinset (nthRoots (↑n) 1))
⊢ Set.F... | /-
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.c... | rw [this] | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ ... | Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
h : IsCyclotomicExtension {n} A B
this : {b | b ^ ↑n = 1} = ↑(Multiset.toFinset (nthRoots (↑n) 1))
⊢ Set.F... | /-
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.c... | exact (nthRoots (↑n) 1).toFinset.finite_toSet | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ ... | Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_2
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
h : IsCyclotomicExtension {n} A B
b : B
hb : b ∈ {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1}
⊢ IsIntegral A b | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ ... | Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_2
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
h : IsCyclotomicExtension {n} A B
b : B
hb : b ^ ↑n = 1
⊢ IsIntegral A b | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩ | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ ... | Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
h : IsCyclotomicExtension {n} A B
b : B
hb : b ^ ↑n = 1
⊢ eval₂ (algebraMap A B) b (X ^ ↑n - 1) = 0 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simp [hb] | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ ... | Mathlib.NumberTheory.Cyclotomic.Basic.308_0.xReI1DeVvechFQU | theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
h₁ : Finite ↑S
h₂ : IsCyclotomicExtension S A B
⊢ Module.Finite A B | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | cases' nonempty_fintype S with h | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
| Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
h₁ : Finite ↑S
h₂ : IsCyclotomicExtension S A B
h : Fintype ↑S
⊢ Module.Finite A B | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | revert h₂ A B | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
| Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro
n : ℕ+
S T : Set ℕ+
K : Type w
L : Type z
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
h₁ : Finite ↑S
h : Fintype ↑S
⊢ ∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B]
[h₂ : IsCyclotomicExtension S A B], Module.Finite A B | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _ | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
| Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine'_1
n : ℕ+
S T : Set ℕ+
K : Type w
L : Type z
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
h₁ : Finite ↑S
h : Fintype ↑S
A : Type u
B : Type v
⊢ ∀ [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B]
[h₂ : IsCyclotomicExtension ∅ A B], Module.Finite A B | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | intro _ _ _ _ _ | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (f... | Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine'_1
n : ℕ+
S T : Set ℕ+
K : Type w
L : Type z
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
h₁ : Finite ↑S
h : Fintype ↑S
A : Type u
B : Type v
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : IsDomain B
h₂✝ : IsCyclotomicExtension ∅ A B
⊢ Module.Finite A B | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩ | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (f... | Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine'_1
n : ℕ+
S T : Set ℕ+
K : Type w
L : Type z
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
h₁ : Finite ↑S
h : Fintype ↑S
A : Type u
B : Type v
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : IsDomain B
h₂✝ : IsCyclotomicExtension ∅ A B
⊢ Submodule.span A ↑{1} = ⊤ | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span] | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (f... | Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine'_2
n✝ : ℕ+
S✝ T : Set ℕ+
K : Type w
L : Type z
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
h₁ : Finite ↑S✝
h : Fintype ↑S✝
n : ℕ+
S : Set ℕ+
x✝¹ : n ∉ S
x✝ : Set.Finite S
H :
∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B]
... | /-
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.c... | intro _ _ _ _ h | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (f... | Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine'_2
n✝ : ℕ+
S✝ T : Set ℕ+
K : Type w
L : Type z
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
h₁ : Finite ↑S✝
h✝ : Fintype ↑S✝
n : ℕ+
S : Set ℕ+
x✝¹ : n ∉ S
x✝ : Set.Finite S
H :
∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B... | /-
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.c... | haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S) | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (f... | Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine'_2
n✝ : ℕ+
S✝ T : Set ℕ+
K : Type w
L : Type z
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
h₁ : Finite ↑S✝
h✝ : Fintype ↑S✝
n : ℕ+
S : Set ℕ+
x✝¹ : n ∉ S
x✝ : Set.Finite S
H :
∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B... | /-
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.c... | haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (f... | Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine'_2
n✝ : ℕ+
S✝ T : Set ℕ+
K : Type w
L : Type z
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
h₁ : Finite ↑S✝
h✝ : Fintype ↑S✝
n : ℕ+
S : Set ℕ+
x✝¹ : n ∉ S
x✝ : Set.Finite S
H :
∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B... | /-
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.c... | have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _ | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (f... | Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
n✝ : ℕ+
S✝ T : Set ℕ+
K : Type w
L : Type z
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
h₁ : Finite ↑S✝
h✝ : Fintype ↑S✝
n : ℕ+
S : Set ℕ+
x✝¹ : n ∉ S
x✝ : Set.Finite S
H :
∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B]
[h₂ : IsCycloto... | /-
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.c... | rw [← union_singleton] at h | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (f... | Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
n✝ : ℕ+
S✝ T : Set ℕ+
K : Type w
L : Type z
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
h₁ : Finite ↑S✝
h✝ : Fintype ↑S✝
n : ℕ+
S : Set ℕ+
x✝¹ : n ∉ S
x✝ : Set.Finite S
H :
∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B]
[h₂ : IsCycloto... | /-
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.c... | letI := @union_right S {n} A B _ _ _ h | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (f... | Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
n✝ : ℕ+
S✝ T : Set ℕ+
K : Type w
L : Type z
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
h₁ : Finite ↑S✝
h✝ : Fintype ↑S✝
n : ℕ+
S : Set ℕ+
x✝¹ : n ∉ S
x✝ : Set.Finite S
H :
∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B]
[h₂ : IsCycloto... | /-
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.c... | exact finite_of_singleton n _ _ | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (f... | Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine'_2
n✝ : ℕ+
S✝ T : Set ℕ+
K : Type w
L : Type z
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
h₁ : Finite ↑S✝
h✝ : Fintype ↑S✝
n : ℕ+
S : Set ℕ+
x✝¹ : n ∉ S
x✝ : Set.Finite S
H :
∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B... | /-
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.c... | exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _ | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (f... | Mathlib.NumberTheory.Cyclotomic.Basic.322_0.xReI1DeVvechFQU | /-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
h : NumberField K
inst✝¹ : Finite ↑S
inst✝ : IsCyclotomicExtension S K L
⊢ FiniteDimensional ℚ L | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | haveI := charZero_of_injective_algebraMap (algebraMap K L).injective | /-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
| Mathlib.NumberTheory.Cyclotomic.Basic.343_0.xReI1DeVvechFQU | /-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
h : NumberField K
inst✝¹ : Finite ↑S
inst✝ : IsCyclotomicExtension S K L
this : CharZero L
⊢ FiniteDimensional ℚ L | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | haveI := IsCyclotomicExtension.finite S K L | /-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective... | Mathlib.NumberTheory.Cyclotomic.Basic.343_0.xReI1DeVvechFQU | /-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
h : NumberField K
inst✝¹ : Finite ↑S
inst✝ : IsCyclotomicExtension S K L
this✝ : CharZero L
this : Module.Finite K L
⊢ FiniteDimensional ℚ L | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | exact Module.Finite.trans K _ | /-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective... | Mathlib.NumberTheory.Cyclotomic.Basic.343_0.xReI1DeVvechFQU | /-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L | Mathlib_NumberTheory_Cyclotomic_Basic |
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
ζ : B
n : ℕ+
hζ : IsPrimitiveRoot ζ ↑n
⊢ adjoin A (rootSet (cyclotomic (↑n) A) B) = adjoin A {b | ∃ a ∈ {n}, b ^ ↑a = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
| Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} | Mathlib_NumberTheory_Cyclotomic_Basic |
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
ζ : B
n : ℕ+
hζ : IsPrimitiveRoot ζ ↑n
⊢ adjoin A (rootSet (cyclotomic (↑n) A) B) = adjoin A {b | b ^ ↑n = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
| Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_1
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
ζ : B
n : ℕ+
hζ : IsPrimitiveRoot ζ ↑n
x : B
hx : x ∈ rootSet (cyclotomic (↑n) A) B
⊢ x ∈ {b | b ^ ↑n = 1... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rw [mem_rootSet'] at hx | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisy... | Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_1
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
ζ : B
n : ℕ+
hζ : IsPrimitiveRoot ζ ↑n
x : B
hx : map (algebraMap A B) (cyclotomic (↑n) A) ≠ 0 ∧ (aeval x... | /-
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.c... | simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisy... | Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_1
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
ζ : B
n : ℕ+
hζ : IsPrimitiveRoot ζ ↑n
x : B
hx : map (algebraMap A B) (cyclotomic (↑n) A) ≠ 0 ∧ (aeval x... | /-
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.c... | rw [isRoot_of_unity_iff n.pos] | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisy... | Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_1
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
ζ : B
n : ℕ+
hζ : IsPrimitiveRoot ζ ↑n
x : B
hx : map (algebraMap A B) (cyclotomic (↑n) A) ≠ 0 ∧ (aeval x... | /-
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.c... | refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisy... | Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_1
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
ζ : B
n : ℕ+
hζ : IsPrimitiveRoot ζ ↑n
x : B
hx : map (algebraMap A B) (cyclotomic (↑n) A) ≠ 0 ∧ (aeval x... | /-
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.c... | rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def] | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisy... | Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_1
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
ζ : B
n : ℕ+
hζ : IsPrimitiveRoot ζ ↑n
x : B
hx : map (algebraMap A B) (cyclotomic (↑n) A) ≠ 0 ∧ (aeval x... | /-
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.c... | exact hx.2 | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisy... | Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_2
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
ζ : B
n : ℕ+
hζ : IsPrimitiveRoot ζ ↑n
x : B
hx : x ∈ {b | b ^ ↑n = 1}
⊢ x ∈ ↑(adjoin A (rootSet (cycloto... | /-
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.c... | simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisy... | Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_2
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
ζ : B
n : ℕ+
hζ : IsPrimitiveRoot ζ ↑n
x : B
hx : x ^ ↑n = 1
⊢ x ∈ ↑(adjoin A (rootSet (cyclotomic (↑n) A... | /-
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.c... | obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisy... | Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_2.intro.intro
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
ζ : B
n : ℕ+
hζ : IsPrimitiveRoot ζ ↑n
i : ℕ
left✝ : i < ↑n
hx : (ζ ^ i) ^ ↑n = 1
⊢ ζ ^ i ∈ ↑... | /-
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.c... | refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisy... | Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_2.intro.intro
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
ζ : B
n : ℕ+
hζ : IsPrimitiveRoot ζ ↑n
i : ℕ
left✝ : i < ↑n
hx : (ζ ^ i) ^ ↑n = 1
⊢ ζ ∈ rootS... | /-
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.c... | rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisy... | Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_2.intro.intro
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsDomain B
ζ : B
n : ℕ+
hζ : IsPrimitiveRoot ζ ↑n
i : ℕ
left✝ : i < ↑n
hx : (ζ ^ i) ^ ↑n = 1
⊢ cyclotomi... | /-
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.c... | refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩ | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisy... | Mathlib.NumberTheory.Cyclotomic.Basic.370_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} | Mathlib_NumberTheory_Cyclotomic_Basic |
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
n : ℕ+
inst✝ : IsDomain B
ζ : B
hζ : IsPrimitiveRoot ζ ↑n
⊢ adjoin A (rootSet (cyclotomic (↑n) A) B) = adjoin A {ζ} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _) | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
| Mathlib.NumberTheory.Cyclotomic.Basic.389_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_1
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
n : ℕ+
inst✝ : IsDomain B
ζ : B
hζ : IsPrimitiveRoot ζ ↑n
x : B
hx : x ∈ rootSet (cyclotomic (↑n) A) B
⊢ x ∈ ↑(adjoin A {ζ}) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | suffices hx : x ^ n.1 = 1 | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· | Mathlib.NumberTheory.Cyclotomic.Basic.389_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_1
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
n : ℕ+
inst✝ : IsDomain B
ζ : B
hζ : IsPrimitiveRoot ζ ↑n
x : B
hx✝ : x ∈ rootSet (cyclotomic (↑n) A) B
hx : x ^ ↑n = 1
⊢ x ... | /-
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.c... | obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
| Mathlib.NumberTheory.Cyclotomic.Basic.389_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_1.intro.intro
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
n : ℕ+
inst✝ : IsDomain B
ζ : B
hζ : IsPrimitiveRoot ζ ↑n
i : ℕ
left✝ : i < ↑n
hx✝ : ζ ^ i ∈ rootSet (cyclotomic... | /-
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.c... | exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _) | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_po... | Mathlib.NumberTheory.Cyclotomic.Basic.389_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} | Mathlib_NumberTheory_Cyclotomic_Basic |
case hx
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
n : ℕ+
inst✝ : IsDomain B
ζ : B
hζ : IsPrimitiveRoot ζ ↑n
x : B
hx : x ∈ rootSet (cyclotomic (↑n) A) B
⊢ x ^ ↑n = 1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' (isRoot_of_unity_iff n.pos B).2 _ | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_po... | Mathlib.NumberTheory.Cyclotomic.Basic.389_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} | Mathlib_NumberTheory_Cyclotomic_Basic |
case hx
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
n : ℕ+
inst✝ : IsDomain B
ζ : B
hζ : IsPrimitiveRoot ζ ↑n
x : B
hx : x ∈ rootSet (cyclotomic (↑n) A) B
⊢ ∃ i ∈ Nat.divisors ↑n, IsR... | /-
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.c... | refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩ | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_po... | Mathlib.NumberTheory.Cyclotomic.Basic.389_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} | Mathlib_NumberTheory_Cyclotomic_Basic |
case hx
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
n : ℕ+
inst✝ : IsDomain B
ζ : B
hζ : IsPrimitiveRoot ζ ↑n
x : B
hx : x ∈ rootSet (cyclotomic (↑n) A) B
⊢ IsRoot (cyclotomic (↑n) B)... | /-
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.c... | rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_po... | Mathlib.NumberTheory.Cyclotomic.Basic.389_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} | Mathlib_NumberTheory_Cyclotomic_Basic |
case hx
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
n : ℕ+
inst✝ : IsDomain B
ζ : B
hζ : IsPrimitiveRoot ζ ↑n
x : B
hx : cyclotomic (↑n) B ≠ 0 ∧ IsRoot (cyclotomic (↑n) B) x
⊢ IsRoot ... | /-
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.c... | exact hx.2 | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_po... | Mathlib.NumberTheory.Cyclotomic.Basic.389_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_2
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
n : ℕ+
inst✝ : IsDomain B
ζ : B
hζ : IsPrimitiveRoot ζ ↑n
x : B
hx : x ∈ {ζ}
⊢ x ∈ rootSet (cyclotomic (↑n) A) B | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_po... | Mathlib.NumberTheory.Cyclotomic.Basic.389_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_2
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
n : ℕ+
inst✝ : IsDomain B
ζ : B
hζ : IsPrimitiveRoot ζ ↑n
x : B
hx : x = ζ
⊢ x ∈ rootSet (cyclotomic (↑n) A) B | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_po... | Mathlib.NumberTheory.Cyclotomic.Basic.389_0.xReI1DeVvechFQU | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} | Mathlib_NumberTheory_Cyclotomic_Basic |
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
n : ℕ+
inst✝ : IsDomain B
h : IsCyclotomicExtension {n} A B
ζ : B
hζ : IsPrimitiveRoot ζ ↑n
⊢ adjoin A {ζ} = ⊤ | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2 | theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
| Mathlib.NumberTheory.Cyclotomic.Basic.404_0.xReI1DeVvechFQU | theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ | Mathlib_NumberTheory_Cyclotomic_Basic |
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
n : ℕ+
inst✝ : IsDomain B
h : IsCyclotomicExtension {n} A B
ζ : B
hζ : IsPrimitiveRoot ζ ↑n
⊢ adjoin A {ζ} = ⊤ | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ] | theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
| Mathlib.NumberTheory.Cyclotomic.Basic.404_0.xReI1DeVvechFQU | theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ | Mathlib_NumberTheory_Cyclotomic_Basic |
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
n : ℕ+
inst✝ : IsDomain B
h : IsCyclotomicExtension {n} A B
ζ : B
hζ : IsPrimitiveRoot ζ ↑n
⊢ adjoin A (rootSet (cyclotomic (↑n) A) B) = ⊤ | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ] | theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
| Mathlib.NumberTheory.Cyclotomic.Basic.404_0.xReI1DeVvechFQU | theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ | Mathlib_NumberTheory_Cyclotomic_Basic |
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
n : ℕ+
inst✝ : IsDomain B
h : IsCyclotomicExtension {n} A B
ζ : B
hζ : IsPrimitiveRoot ζ ↑n
⊢ adjoin A {b | ∃ a ∈ {n}, b ^ ↑a = 1} = ⊤ | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | exact ((iff_adjoin_eq_top {n} A B).mp h).2 | theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
| Mathlib.NumberTheory.Cyclotomic.Basic.404_0.xReI1DeVvechFQU | theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ | Mathlib_NumberTheory_Cyclotomic_Basic |
n✝¹ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
ζ : B
n : ℕ+
h : IsPrimitiveRoot ζ ↑n
n✝ : ℕ+
hi : n✝ ∈ {n}
⊢ ∃ r, IsPrimitiveRoot r ↑n✝ | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rw [Set.mem_singleton_iff] at hi | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
| Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) | Mathlib_NumberTheory_Cyclotomic_Basic |
n✝¹ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
ζ : B
n : ℕ+
h : IsPrimitiveRoot ζ ↑n
n✝ : ℕ+
hi : n✝ = n
⊢ ∃ r, IsPrimitiveRoot r ↑n✝ | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩ | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
| Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) | Mathlib_NumberTheory_Cyclotomic_Basic |
n✝¹ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
ζ : B
n : ℕ+
h : IsPrimitiveRoot ζ ↑n
n✝ : ℕ+
hi : n✝ = n
⊢ IsPrimitiveRoot { val := ζ, property := (_ : ζ ∈ ↑(adjoin A {ζ})) } ↑n✝ | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi] | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
| Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) | Mathlib_NumberTheory_Cyclotomic_Basic |
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
ζ : B
n : ℕ+
h : IsPrimitiveRoot ζ ↑n
x : ↥(adjoin A {ζ})
⊢ x ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_) | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPri... | Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_1
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
ζ : B
n : ℕ+
h : IsPrimitiveRoot ζ ↑n
x : ↥(adjoin A {ζ})
b : B
hb : b ∈ {ζ}
⊢ { val := b, property := (_ : b ∈ ↑(adjoin A {ζ}... | /-
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.c... | rw [Set.mem_singleton_iff] at hb | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPri... | Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_1
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
ζ : B
n : ℕ+
h : IsPrimitiveRoot ζ ↑n
x : ↥(adjoin A {ζ})
b : B
hb✝ : b ∈ {ζ}
hb : b = ζ
⊢ { val := b, property := (_ : b ∈ ↑(... | /-
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.c... | refine' subset_adjoin _ | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPri... | Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_1
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
ζ : B
n : ℕ+
h : IsPrimitiveRoot ζ ↑n
x : ↥(adjoin A {ζ})
b : B
hb✝ : b ∈ {ζ}
hb : b = ζ
⊢ { val := b, property := (_ : b ∈ ↑(... | /-
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.c... | simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb] | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPri... | Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_1
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
ζ : B
n : ℕ+
h : IsPrimitiveRoot ζ ↑n
x : ↥(adjoin A {ζ})
b : B
hb✝ : b ∈ {ζ}
hb : b = ζ
⊢ { val := ζ, property := (_ : (fun x... | /-
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.c... | rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPri... | Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_1
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
ζ : B
n : ℕ+
h : IsPrimitiveRoot ζ ↑n
x : ↥(adjoin A {ζ})
b : B
hb✝ : b ∈ {ζ}
hb : b = ζ
⊢ ζ ^ ↑n = 1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1 | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPri... | Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_2
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
ζ : B
n : ℕ+
h : IsPrimitiveRoot ζ ↑n
x : ↥(adjoin A {ζ})
a : A
⊢ (algebraMap A ↥(adjoin A {ζ})) a ∈ adjoin A {b | ∃ n_1 ∈ {n}... | /-
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.c... | exact Subalgebra.algebraMap_mem _ _ | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPri... | Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_3
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
ζ : B
n : ℕ+
h : IsPrimitiveRoot ζ ↑n
x b₁ b₂ : ↥(adjoin A {ζ})
hb₁ : b₁ ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1}
hb₂ : b₂ ∈... | /-
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.c... | exact Subalgebra.add_mem _ hb₁ hb₂ | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPri... | Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_4
n✝ : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
ζ : B
n : ℕ+
h : IsPrimitiveRoot ζ ↑n
x b₁ b₂ : ↥(adjoin A {ζ})
hb₁ : b₁ ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1}
hb₂ : b₂ ∈... | /-
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.c... | exact Subalgebra.mul_mem _ hb₁ hb₂ | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPri... | Mathlib.NumberTheory.Cyclotomic.Basic.414_0.xReI1DeVvechFQU | theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
H : IsCyclotomicExtension S K L
hS : n ∈ S
⊢ Splits (algebraMap K L) (X ^ ↑n - 1) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X] | /-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.441_0.xReI1DeVvechFQU | /-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
H : IsCyclotomicExtension S K L
hS : n ∈ S
⊢ Splits (RingHom.id L) (X ^ ↑n - 1) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS | /-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
| Mathlib.NumberTheory.Cyclotomic.Basic.441_0.xReI1DeVvechFQU | /-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
H : IsCyclotomicExtension S K L
hS : n ∈ S
z : L
hz : IsPrimitiveRoot z ↑n
⊢ Splits (RingHom.id L) (X ^ ↑n - 1) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | exact X_pow_sub_one_splits hz | /-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ :... | Mathlib.NumberTheory.Cyclotomic.Basic.441_0.xReI1DeVvechFQU | /-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension S K L
hS : n ∈ S
⊢ Splits (algebraMap K L) (cyclotomic (↑n) K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _ | /-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.451_0.xReI1DeVvechFQU | /-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension S K L
hS : n ∈ S
⊢ cyclotomic (↑n) K ∣ X ^ ↑n - C 1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K | /-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
| Mathlib.NumberTheory.Cyclotomic.Basic.451_0.xReI1DeVvechFQU | /-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
case h
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension S K L
hS : n ∈ S
⊢ X ^ ↑n - C 1 = cyclotomic (↑n) K * ∏ i in Nat.properDivisors ↑n, cyclotomic i K | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one] | /-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ... | Mathlib.NumberTheory.Cyclotomic.Basic.451_0.xReI1DeVvechFQU | /-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {n} K L
⊢ adjoin K (rootSet (X ^ ↑n - 1) L) = ⊤ | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2] | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
| Mathlib.NumberTheory.Cyclotomic.Basic.465_0.xReI1DeVvechFQU | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {n} K L
⊢ adjoin K (rootSet (X ^ ↑n - 1) L) = adjoin K {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | congr | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
... | Mathlib.NumberTheory.Cyclotomic.Basic.465_0.xReI1DeVvechFQU | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) | Mathlib_NumberTheory_Cyclotomic_Basic |
case e_s
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {n} K L
⊢ rootSet (X ^ ↑n - 1) L = {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | refine' Set.ext fun x => _ | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
... | Mathlib.NumberTheory.Cyclotomic.Basic.465_0.xReI1DeVvechFQU | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) | Mathlib_NumberTheory_Cyclotomic_Basic |
case e_s
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {n} K L
x : L
⊢ x ∈ rootSet (X ^ ↑n - 1) L ↔ x ∈ {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub] | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
... | Mathlib.NumberTheory.Cyclotomic.Basic.465_0.xReI1DeVvechFQU | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) | Mathlib_NumberTheory_Cyclotomic_Basic |
case e_s
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {n} K L
x : L
⊢ x ∈ rootSet (X ^ ↑n - 1) L ↔ x ^ ↑n = 1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one] | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
... | Mathlib.NumberTheory.Cyclotomic.Basic.465_0.xReI1DeVvechFQU | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) | Mathlib_NumberTheory_Cyclotomic_Basic |
case e_s
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {n} K L
x : L
⊢ x ^ ↑n = 1 → X ^ ↑n - 1 ≠ 0 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
... | Mathlib.NumberTheory.Cyclotomic.Basic.465_0.xReI1DeVvechFQU | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {n} K L
⊢ adjoin K (rootSet (cyclotomic (↑n) K) L) = ⊤ | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2] | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
| Mathlib.NumberTheory.Cyclotomic.Basic.497_0.xReI1DeVvechFQU | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {n} K L
⊢ adjoin K (rootSet (cyclotomic (↑n) K) L) = adjoin K {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.c... | letI := Classical.decEq L | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
... | Mathlib.NumberTheory.Cyclotomic.Basic.497_0.xReI1DeVvechFQU | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
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