mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.DedekindDomain.LinearDisjoint

Goal state before:
pp: A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst✝³⁴ : CommRing A
inst✝³³ : Field K
inst✝³² : Algebra A K
inst✝³¹ : IsFractionRing A K
inst✝³⁰ : CommRing B
inst✝²⁹ : Field L
inst✝²⁸ : Algebra B L
inst✝²⁷ : Algebra A L
inst✝²⁶ : Algebra K L
inst✝²⁵ : FiniteDimensional K L
inst✝²⁴ : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst✝²³ : CommRing R₁
inst✝²² : CommRing R₂
inst✝²¹ : IsDomain R₁
inst✝²⁰ : Algebra A R₁
inst✝¹⁹ : Algebra A R₂
inst✝¹⁸ : Algebra R₁ B
inst✝¹⁷ : Algebra R₂ B
inst✝¹⁶ : Algebra R₁ L
inst✝¹⁵ : Algebra R₂ L
inst✝¹⁴ : IsScalarTower A R₁ L
inst✝¹³ : IsScalarTower R₁ B L
inst✝¹² : IsScalarTower R₂ B L
inst✝¹¹ : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst✝¹⁰ : Algebra R₁ ↥F₁
inst✝⁹ : Algebra R₂ ↥F₂
inst✝⁸ : IsTorsionFree R₁ ↥F₁
inst✝⁷ : IsScalarTower A (↥F₂) L
inst✝⁶ : IsScalarTower A R₂ ↥F₂
inst✝⁵ : IsScalarTower R₁ (↥F₁) L
inst✝⁴ : IsScalarTower R₂ (↥F₂) L
inst✝³ : Algebra.IsSeparable K ↥F₂
inst✝² : Algebra.IsSeparable (↥F₁) L
inst✝¹ : Free A R₂
inst✝ : IsLocalization (algebraMapSubmonoid R₂ A⁰) ↥F₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁ ⊔ F₂ = ⊤
x : L
hx : x ∈ restrictScalars R₁ (traceDual R₁ (↥F₁) 1)
h₂' : F₁.toSubalgebra ⊔ F₂.toSubalgebra = ⊤
b₂ : Basis (Free.ChooseBasisIndex A R₂) K ↥F₂ := Basis.localizationLocalization K A⁰ (↥F₂) (Free.chooseBasis A R₂)
B₁ : Basis (Free.ChooseBasisIndex A R₂) (↥F₁) L := h₁.basisOfBasisRight h₂' b₂
⊢ x ∈ span R₁ (Set.range ⇑B₁.traceDual)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p k : ℕ
hp : Fact (Nat.Prime p)
⊢ IsIntegralClosure (CyclotomicRing (p ^ k) ℤ ℚ) ℤ (CyclotomicField (p ^ k) ℚ)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
⊢ IsIntegralClosure (CyclotomicRing p ℤ ℚ) ℤ (CyclotomicField p ℚ)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: K : Type u
inst✝¹ : Field K
ζ : K
inst✝ : NumberField K
k : ℕ
hk : 1 < k
hζ : IsPrimitiveRoot ζ k
⊢ Finite (𝓞 K ⧸ Ideal.span {hζ.toInteger - 1})

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: R : Type u
K : Type v
L : Type z
p : R
inst✝⁹ : CommRing R
inst✝⁸ : Field K
inst✝⁷ : Field L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : Algebra R K
inst✝³ : IsScalarTower R K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
Q : R[X]
hQ : (aeval B.gen) Q = p • z
hzint : IsIntegral R z
hei : (minpoly R B.gen).IsEisensteinAt (R ∙ p)
⊢ p ∣ Q.coeff 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : NumberField K
k : ℕ
inst✝ : NeZero k
hζ : IsPrimitiveRoot ζ k
⊢ Nat.card (𝓞 K ⧸ Ideal.span {hζ.toInteger - 1}) = ((Algebra.norm ℤ) (hζ.toInteger - 1)).natAbs

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: K : Type u
inst✝¹ : Field K
ζ : K
k : ℕ
inst✝ : NeZero k
hζ : IsPrimitiveRoot ζ k
⊢ IsPrimitiveRoot ((algebraMap (𝓞 K) K) hζ.toInteger) k

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p k : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p ^ k} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ k)
⊢ hζ.integralPowerBasisOfPrimePow.dim = φ (p ^ k)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p k n : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ k} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ k)
⊢ ℤ[⟨ζ - 1, ⋯⟩] = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ k} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ k)
⊢ hζ.subOneIntegralPowerBasisOfPrimePow.gen = ⟨ζ - 1, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
h : hζ.toInteger - 1 = 0
⊢ k + 1 ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁴ : CommSemiring A
inst✝³ : Ring B
inst✝² : Algebra A B
inst✝¹ : IsDomain A
inst✝ : Module.IsTorsionFree A B
Q : A[X]
p : A
x z : B
hp : p ≠ 0
hQ : ∀ i ∈ range (Q.natDegree + 1), p ∣ Q.coeff i
hz : (aeval x) Q = p • z
⊢ z ∈ A[x]

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.LinearDisjoint

Goal state before:
pp: A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst✝³⁴ : CommRing A
inst✝³³ : Field K
inst✝³² : Algebra A K
inst✝³¹ : IsFractionRing A K
inst✝³⁰ : CommRing B
inst✝²⁹ : Field L
inst✝²⁸ : Algebra B L
inst✝²⁷ : Algebra A L
inst✝²⁶ : Algebra K L
inst✝²⁵ : FiniteDimensional K L
inst✝²⁴ : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst✝²³ : CommRing R₁
inst✝²² : CommRing R₂
inst✝²¹ : IsDomain R₁
inst✝²⁰ : Algebra A R₁
inst✝¹⁹ : Algebra A R₂
inst✝¹⁸ : Algebra R₁ B
inst✝¹⁷ : Algebra R₂ B
inst✝¹⁶ : Algebra R₁ L
inst✝¹⁵ : Algebra R₂ L
inst✝¹⁴ : IsScalarTower A R₁ L
inst✝¹³ : IsScalarTower R₁ B L
inst✝¹² : IsScalarTower R₂ B L
inst✝¹¹ : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst✝¹⁰ : Algebra R₁ ↥F₁
inst✝⁹ : Algebra R₂ ↥F₂
inst✝⁸ : IsTorsionFree R₁ ↥F₁
inst✝⁷ : IsScalarTower A (↥F₂) L
inst✝⁶ : IsScalarTower A R₂ ↥F₂
inst✝⁵ : IsScalarTower R₁ (↥F₁) L
inst✝⁴ : IsScalarTower R₂ (↥F₂) L
inst✝³ : Algebra.IsSeparable K ↥F₂
inst✝² : Algebra.IsSeparable (↥F₁) L
inst✝¹ : Free A R₂
inst✝ : IsLocalization (algebraMapSubmonoid R₂ A⁰) ↥F₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁ ⊔ F₂ = ⊤
x : L
hx : x ∈ restrictScalars R₁ (traceDual R₁ (↥F₁) 1)
h₂' : F₁.toSubalgebra ⊔ F₂.toSubalgebra = ⊤
b₂ : Basis (Free.ChooseBasisIndex A R₂) K ↥F₂ := Basis.localizationLocalization K A⁰ (↥F₂) (Free.chooseBasis A R₂)
B₁ : Basis (Free.ChooseBasisIndex A R₂) (↥F₁) L := h₁.basisOfBasisRight h₂' b₂
h_main : x ∈ span R₁ (Set.range ⇑B₁.traceDual)
⊢ Set.range ⇑B₁.traceDual = Set.range (⇑(IsScalarTower.toAlgHom A (↥F₂) L) ∘ ⇑b₂.traceDual)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
hodd : p ≠ 2
⊢ Prime (hζ.toInteger - 1)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
hodd : p ≠ 2
⊢ Prime (hζ.toInteger - 1)

Tactic:
by
  let := IsCyclotomicExtension.numberField {p ^ (k + 1)} ℚ K
  refine Ideal.prime_of_irreducible_absNorm_span (fun h ↦ ?_) ?_
  · apply hζ.pow_ne_one_of_pos_of_lt one_ne_zero (one_lt_pow₀ hp.out.one_lt (by simp))
    rw [sub_eq_zero] at h
    simpa using congrArg (algebraMap _ K) h
  rw [Nat.irreducible_iff_prime, Ideal.absNorm_span_singleton, ← Nat.prime_iff,
    ← Int.prime_iff_natAbs_prime]
  convert! Nat.prime_iff_prime_int.1 hp.out
  apply RingHom.injective_int (algebraMap ℤ ℚ)
  rw [← Algebra.norm_localization (Sₘ := K) ℤ (nonZeroDivisors ℤ)]
  simp only [algebraMap_int_eq, map_natCast]
  exact hζ.norm_sub_one_of_prime_ne_two (Polynomial.cyclotomic.irreducible_rat (NeZero.pos _)) hodd

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: k : ℕ
K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (2 ^ (k + 1))
this : NumberField K
h : hζ.toInteger - 1 = 0
⊢ 1 < 2

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: R : Type u
K : Type v
L : Type z
p : R
inst✝⁹ : CommRing R
inst✝⁸ : Field K
inst✝⁷ : Field L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : Algebra R K
inst✝³ : IsScalarTower R K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝ : Module.Finite K L
P : R[X] := minpoly R B.gen
hei : P.IsEisensteinAt (R ∙ p)
hndiv : ¬p ^ 2 ∣ P.coeff 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
this : Module.IsTorsionFree R L
Q₁ : R[X]
hQ : (aeval B.gen).toRingHom Q₁ = p • z
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
⊢ (aeval B.gen) Q = p • z

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: k : ℕ
K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (2 ^ (k + 1))
this : NumberField K
h : hζ.toInteger - 1 = 0
⊢ k + 1 ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
this : NumberField K
inst✝ : IsCyclotomicExtension {2 ^ (0 + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (2 ^ (0 + 1))
⊢ 0 < 2 ^ (0 + 1)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: R : Type u
K : Type v
L : Type z
p : R
inst✝⁹ : CommRing R
inst✝⁸ : Field K
inst✝⁷ : Field L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : Algebra R K
inst✝³ : IsScalarTower R K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝ : Module.Finite K L
P : R[X] := minpoly R B.gen
hei : P.IsEisensteinAt (R ∙ p)
hndiv : ¬p ^ 2 ∣ P.coeff 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
this : Module.IsTorsionFree R L
Q₁ : R[X]
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
hQzero : Q = 0
H : (algebraMap R L) p = 0
⊢ Function.Injective ⇑(algebraMap R L)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
this : NumberField K
n✝ : ℕ
inst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (2 ^ (n✝ + 1 + 1))
⊢ 0 < 2 ^ (n✝ + 1 + 1)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: k : ℕ
K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (2 ^ (k + 1))
⊢ Prime (hζ.toInteger - 1)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
⊢ Prime (hζ.toInteger - 1)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
h : IsCyclotomicExtension {p} ℚ K
hζ : IsPrimitiveRoot ζ p
⊢ IsPrimitiveRoot ?m.30 (?m.25 ^ (0 + 1))

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
h : IsCyclotomicExtension {p} ℚ K
hζ : IsPrimitiveRoot ζ p
⊢ Prime (hζ.toInteger - 1)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
⊢ Prime hζ.subOneIntegralPowerBasisOfPrimePow.gen

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
n : ℕ
inst✝ : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot ζ n
h₁ : 2 < n
h₂ : ∀ {p : ℕ}, Nat.Prime p → ∀ (k : ℕ), p ^ k ≠ n
⊢ have this := ⋯;
  (Algebra.norm ℤ) (hζ.toInteger - 1) = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.LinearDisjoint

Goal state before:
pp: A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst✝³⁴ : CommRing A
inst✝³³ : Field K
inst✝³² : Algebra A K
inst✝³¹ : IsFractionRing A K
inst✝³⁰ : CommRing B
inst✝²⁹ : Field L
inst✝²⁸ : Algebra B L
inst✝²⁷ : Algebra A L
inst✝²⁶ : Algebra K L
inst✝²⁵ : FiniteDimensional K L
inst✝²⁴ : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst✝²³ : CommRing R₁
inst✝²² : CommRing R₂
inst✝²¹ : IsDomain R₁
inst✝²⁰ : Algebra A R₁
inst✝¹⁹ : Algebra A R₂
inst✝¹⁸ : Algebra R₁ B
inst✝¹⁷ : Algebra R₂ B
inst✝¹⁶ : Algebra R₁ L
inst✝¹⁵ : Algebra R₂ L
inst✝¹⁴ : IsScalarTower A R₁ L
inst✝¹³ : IsScalarTower R₁ B L
inst✝¹² : IsScalarTower R₂ B L
inst✝¹¹ : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst✝¹⁰ : Algebra R₁ ↥F₁
inst✝⁹ : Algebra R₂ ↥F₂
inst✝⁸ : IsTorsionFree R₁ ↥F₁
inst✝⁷ : IsScalarTower A (↥F₂) L
inst✝⁶ : IsScalarTower A R₂ ↥F₂
inst✝⁵ : IsScalarTower R₁ (↥F₁) L
inst✝⁴ : IsScalarTower R₂ (↥F₂) L
inst✝³ : Algebra.IsSeparable K ↥F₂
inst✝² : Algebra.IsSeparable (↥F₁) L
inst✝¹ : Free A R₂
inst✝ : IsLocalization (algebraMapSubmonoid R₂ A⁰) ↥F₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁ ⊔ F₂ = ⊤
x : L
hx : x ∈ restrictScalars R₁ (traceDual R₁ (↥F₁) 1)
h₂' : F₁.toSubalgebra ⊔ F₂.toSubalgebra = ⊤
b₂ : Basis (Free.ChooseBasisIndex A R₂) K ↥F₂ := Basis.localizationLocalization K A⁰ (↥F₂) (Free.chooseBasis A R₂)
B₁ : Basis (Free.ChooseBasisIndex A R₂) (↥F₁) L := h₁.basisOfBasisRight h₂' b₂
h_main : x ∈ span R₁ ↑(map (IsScalarTower.toAlgHom A (↥F₂) L).toLinearMap (span A (Set.range ⇑b₂.traceDual)))
h : Set.range ⇑B₁.traceDual = Set.range (⇑(IsScalarTower.toAlgHom A (↥F₂) L) ∘ ⇑b₂.traceDual)
⊢ restrictScalars A 1 = span A (Set.range ⇑b₂)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
s : ℕ
hs : s ≤ k
htwo : p ^ (k - s + 1) ≠ 2
⊢ (Algebra.norm ℤ) (hζ.toInteger ^ p ^ s - 1) = ↑p ^ p ^ s

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: k : ℕ
K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (2 ^ (k + 1))
this : NumberField K
⊢ 0 < 2

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: k : ℕ
K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (2 ^ (k + 1))
⊢ (Algebra.norm ℤ) (hζ.toInteger ^ 2 ^ k - 1) = (-2) ^ 2 ^ k

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
s : ℕ
hs : s ≤ k
hodd : p ≠ 2
⊢ (Algebra.norm ℤ) (hζ.toInteger ^ p ^ s - 1) = ↑p ^ p ^ s

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.LinearDisjoint

Goal state before:
pp: A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst✝³⁴ : CommRing A
inst✝³³ : Field K
inst✝³² : Algebra A K
inst✝³¹ : IsFractionRing A K
inst✝³⁰ : CommRing B
inst✝²⁹ : Field L
inst✝²⁸ : Algebra B L
inst✝²⁷ : Algebra A L
inst✝²⁶ : Algebra K L
inst✝²⁵ : FiniteDimensional K L
inst✝²⁴ : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst✝²³ : CommRing R₁
inst✝²² : CommRing R₂
inst✝²¹ : IsDomain R₁
inst✝²⁰ : Algebra A R₁
inst✝¹⁹ : Algebra A R₂
inst✝¹⁸ : Algebra R₁ B
inst✝¹⁷ : Algebra R₂ B
inst✝¹⁶ : Algebra R₁ L
inst✝¹⁵ : Algebra R₂ L
inst✝¹⁴ : IsScalarTower A R₁ L
inst✝¹³ : IsScalarTower R₁ B L
inst✝¹² : IsScalarTower R₂ B L
inst✝¹¹ : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst✝¹⁰ : Algebra R₁ ↥F₁
inst✝⁹ : Algebra R₂ ↥F₂
inst✝⁸ : IsTorsionFree R₁ ↥F₁
inst✝⁷ : IsScalarTower A (↥F₂) L
inst✝⁶ : IsScalarTower A R₂ ↥F₂
inst✝⁵ : IsScalarTower R₁ (↥F₁) L
inst✝⁴ : IsScalarTower R₂ (↥F₂) L
inst✝³ : Algebra.IsSeparable K ↥F₂
inst✝² : Algebra.IsSeparable (↥F₁) L
inst✝¹ : Free A R₂
inst✝ : IsLocalization (algebraMapSubmonoid R₂ A⁰) ↥F₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁ ⊔ F₂ = ⊤
⊢ restrictScalars R₁ (traceDual R₁ (↥F₁) 1) ≤ span R₁ (⇑(algebraMap (↥F₂) L) '' ↑(traceDual A K 1))

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: k : ℕ
K : Type u_1
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {2 ^ (k + 2)} ℚ K
hζ : IsPrimitiveRoot ζ (2 ^ (k + 2))
⊢ (Algebra.norm ℤ) (hζ.toInteger - 1) = 2

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
hodd : p ≠ 2
⊢ (Algebra.norm ℤ) (hζ.toInteger - 1) = ↑p

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {2} ℚ K
hζ : IsPrimitiveRoot ζ 2
⊢ (Algebra.norm ℤ) (hζ.toInteger - 1) = -2

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p} ℚ K
hζ : IsPrimitiveRoot ζ p
h : p ≠ 2
⊢ IsCyclotomicExtension {p ^ (0 + 1)} ℚ K

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.LinearDisjoint

Goal state before:
pp: A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst✝⁴⁶ : CommRing A
inst✝⁴⁵ : Field K
inst✝⁴⁴ : Algebra A K
inst✝⁴³ : IsFractionRing A K
inst✝⁴² : CommRing B
inst✝⁴¹ : Field L
inst✝⁴⁰ : Algebra B L
inst✝³⁹ : Algebra A L
inst✝³⁸ : Algebra K L
inst✝³⁷ : FiniteDimensional K L
inst✝³⁶ : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst✝³⁵ : CommRing R₁
inst✝³⁴ : CommRing R₂
inst✝³³ : IsDomain R₁
inst✝³² : Algebra A R₁
inst✝³¹ : Algebra A R₂
inst✝³⁰ : Algebra R₁ B
inst✝²⁹ : Algebra R₂ B
inst✝²⁸ : Algebra R₁ L
inst✝²⁷ : Algebra R₂ L
inst✝²⁶ : IsScalarTower A R₁ L
inst✝²⁵ : IsScalarTower R₁ B L
inst✝²⁴ : IsScalarTower R₂ B L
inst✝²³ : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst✝²² : Algebra R₁ ↥F₁
inst✝²¹ : Algebra R₂ ↥F₂
inst✝²⁰ : IsTorsionFree R₁ ↥F₁
inst✝¹⁹ : IsScalarTower A (↥F₂) L
inst✝¹⁸ : IsScalarTower A R₂ ↥F₂
inst✝¹⁷ : IsScalarTower R₁ (↥F₁) L
inst✝¹⁶ : IsScalarTower R₂ (↥F₂) L
inst✝¹⁵ : Algebra.IsSeparable K ↥F₂
inst✝¹⁴ : Algebra.IsSeparable (↥F₁) L
inst✝¹³ : IsDomain A
inst✝¹² : IsDedekindDomain B
inst✝¹¹ : IsDedekindDomain R₁
inst✝¹⁰ : IsDedekindDomain R₂
inst✝⁹ : IsFractionRing B L
inst✝⁸ : IsFractionRing R₁ ↥F₁
inst✝⁷ : IsFractionRing R₂ ↥F₂
inst✝⁶ : IsIntegrallyClosed A
inst✝⁵ : IsIntegralClosure B R₁ L
inst✝⁴ : IsTorsionFree R₁ B
inst✝³ : IsTorsionFree R₂ B
inst✝² : Algebra.IsIntegral R₂ B
inst✝¹ : Free A R₂
inst✝ : IsLocalization (algebraMapSubmonoid R₂ A⁰) ↥F₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁ ⊔ F₂ = ⊤
⊢ Algebra.IsSeparable (FractionRing A) (FractionRing R₂)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p} ℚ K
hζ : IsPrimitiveRoot ζ p
h : p ≠ 2
this : IsCyclotomicExtension {p ^ (0 + 1)} ℚ K
⊢ IsPrimitiveRoot ζ (p ^ (0 + 1))

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p} ℚ K
hζ : IsPrimitiveRoot ζ p
h : p ≠ 2
⊢ (Algebra.norm ℤ) (hζ.toInteger - 1) = ↑p

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
htwo : p ^ (k + 1) ≠ 2
⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p k : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
hodd : p ≠ 2
⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p} ℚ K
hζ : IsPrimitiveRoot ζ p
hodd : p ≠ 2
⊢ IsCyclotomicExtension {p ^ (0 + 1)} ℚ K

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p} ℚ K
hζ : IsPrimitiveRoot ζ p
hodd : p ≠ 2
this : IsCyclotomicExtension {p ^ (0 + 1)} ℚ K
⊢ IsPrimitiveRoot ζ (p ^ (0 + 1))

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p} ℚ K
hζ : IsPrimitiveRoot ζ p
hodd : p ≠ 2
⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p k : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
htwo : p ^ (k + 1) ≠ 2
n : ℤ
x : 𝓞 K
h : hζ.toInteger - ↑n = ↑p * x
pB : PowerBasis ℤ (𝓞 K) := hζ.integralPowerBasisOfPrimePow
⊢ pB.dim = p ^ k * (p - 1)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: R : Type u
K : Type v
L : Type z
p : R
inst✝⁹ : CommRing R
inst✝⁸ : Field K
inst✝⁷ : Field L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : Algebra R K
inst✝³ : IsScalarTower R K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝ : Module.Finite K L
P : R[X] := minpoly R B.gen
hei : P.IsEisensteinAt (R ∙ p)
hndiv : ¬p ^ 2 ∣ P.coeff 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
this : Module.IsTorsionFree R L
Q₁ : R[X]
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
hQ : (aeval B.gen) Q = p • z
hQzero : ¬Q = 0
j : ℕ
hind : ∀ m ≤ j, m ∈ range (Q.natDegree + 1) → p ∣ Q.coeff m
hj : j + 1 ∈ range (Q.natDegree + 1)
H : j + 1 + 1 ≤ P.natDegree
⊢ Q.natDegree + 1 = j + 1 + (Q.natDegree - j)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p k : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
htwo : p ^ (k + 1) ≠ 2
n : ℤ
x : 𝓞 K
h : hζ.toInteger - ↑n = ↑p * x
pB : PowerBasis ℤ (𝓞 K) := hζ.integralPowerBasisOfPrimePow
hdim : pB.dim = p ^ k * (p - 1)
⊢ p ^ k * (p - 1) ≠ 0

Tactic:
by

Goal state after: