Theorem: Hilbert92ish
Proof state before:
case intro.intro
p : ℕ+
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : NumberField K
k : Type u_2
inst✝¹¹ : Field k
inst✝¹⁰ : NumberField k
hp : (↑p).Prime
G : Type u_3
H✝ : Type u_4
inst✝⁹ : AddCommGroup G
inst✝⁸ : CommGroup H✝
inst✝⁷ : Fintype H✝
hCard : Fintype.card H✝ = ↑p
σ✝ : H✝
hσ✝ : Subgroup.zpowers σ✝ = ⊤
r : ℕ
inst✝⁶ : DistribMulAction H✝ G
inst✝⁵ : Module.Free ℤ G
inst✝⁴ : Module.Finite ℤ G
hf : finrank ℤ G = r * (↑p - 1)
inst✝³ : Algebra k K
inst✝² : IsGalois k K
inst✝¹ : FiniteDimensional k K
inst✝ : IsUnramifiedAtInfinitePlaces k K
hKL : finrank k K = ↑p
σ : K ≃ₐ[k] K
hσ : ∀ (x : K ≃ₐ[k] K), x ∈ Subgroup.zpowers σ
hpodd : ↑p ≠ 2
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
NE : (𝓞 k)ˣ := (Units.map (RingOfIntegers.norm k)) E
hNE : (algebraMap (𝓞 k) k) ↑NE = (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑E)
S :
  systemOfUnits p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1)
hS :
  systemOfUnits.IsFundamental p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1) S
H : Fin (r + 1) → Additive (𝓞 K)ˣ := unitlifts p hp hKL σ hσ S
N : Fin (r + 1) → Additive (𝓞 k)ˣ :=
  fun e => Additive.ofMul ((Units.map (RingOfIntegers.norm k)) (Additive.toMul (H e)))
η : Fin (r + 2) → Additive (𝓞 k)ˣ := Fin.snoc N (Additive.ofMul NE)
a : ℤ
ι : Fin (r + 2) → ℤ
i✝ : Fin (r + 2)
ha : ∑ i : Fin (r + 2), ι i • η i = (a * ↑↑p) • Additive.ofMul ζ
ha' : ¬↑↑p ∣ ι (Fin.last (r + 1))
ha'' : ζ = 1 → i✝ ≠ Fin.last (r + 1)
H2 : Fin (r + 2) → Additive (𝓞 K)ˣ := Fin.snoc H (Additive.ofMul E)
J : (𝓞 K)ˣ := Additive.toMul (∑ i : Fin (r + 2), ι i • H2 i) * (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ζ ^ (-a)
ε : (𝓞 K)ˣ
hε : (algebraMap (𝓞 K) K) ↑J = (algebraMap (𝓞 K) K) ↑ε / σ ((algebraMap (𝓞 K) K) ↑ε)
hε'✝ : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i✝ = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
ε' : (𝓞 k)ˣ
hε' : ε' ^ ↑p = NE
hζ'' : IsPrimitiveRoot ((algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h)) ↑p
NE_p_pow : (algebraMap (𝓞 K) K) ((↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹) ^ ↑p) = (algebraMap (𝓞 K) K) 1
i : ℕ
e : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h) ^ i = ↑(↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹)
⊢ ∃ ε'', E = (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ε''
Tactic:
use ((ζ ^ (p : ℕ) ^ h) ^ i * ε')
Proof state after:
case h
p : ℕ+
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : NumberField K
k : Type u_2
inst✝¹¹ : Field k
inst✝¹⁰ : NumberField k
hp : (↑p).Prime
G : Type u_3
H✝ : Type u_4
inst✝⁹ : AddCommGroup G
inst✝⁸ : CommGroup H✝
inst✝⁷ : Fintype H✝
hCard : Fintype.card H✝ = ↑p
σ✝ : H✝
hσ✝ : Subgroup.zpowers σ✝ = ⊤
r : ℕ
inst✝⁶ : DistribMulAction H✝ G
inst✝⁵ : Module.Free ℤ G
inst✝⁴ : Module.Finite ℤ G
hf : finrank ℤ G = r * (↑p - 1)
inst✝³ : Algebra k K
inst✝² : IsGalois k K
inst✝¹ : FiniteDimensional k K
inst✝ : IsUnramifiedAtInfinitePlaces k K
hKL : finrank k K = ↑p
σ : K ≃ₐ[k] K
hσ : ∀ (x : K ≃ₐ[k] K), x ∈ Subgroup.zpowers σ
hpodd : ↑p ≠ 2
ζ : (𝓞 k)ˣ
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
NE : (𝓞 k)ˣ := (Units.map (RingOfIntegers.norm k)) E
hNE : (algebraMap (𝓞 k) k) ↑NE = (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑E)
S :
  systemOfUnits p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1)
hS :
  systemOfUnits.IsFundamental p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1) S
H : Fin (r + 1) → Additive (𝓞 K)ˣ := unitlifts p hp hKL σ hσ S
N : Fin (r + 1) → Additive (𝓞 k)ˣ :=
  fun e => Additive.ofMul ((Units.map (RingOfIntegers.norm k)) (Additive.toMul (H e)))
η : Fin (r + 2) → Additive (𝓞 k)ˣ := Fin.snoc N (Additive.ofMul NE)
a : ℤ
ι : Fin (r + 2) → ℤ
i✝ : Fin (r + 2)
ha : ∑ i : Fin (r + 2), ι i • η i = (a * ↑↑p) • Additive.ofMul ζ
ha' : ¬↑↑p ∣ ι (Fin.last (r + 1))
ha'' : ζ = 1 → i✝ ≠ Fin.last (r + 1)
H2 : Fin (r + 2) → Additive (𝓞 K)ˣ := Fin.snoc H (Additive.ofMul E)
J : (𝓞 K)ˣ := Additive.toMul (∑ i : Fin (r + 2), ι i • H2 i) * (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ζ ^ (-a)
ε : (𝓞 K)ˣ
hε : (algebraMap (𝓞 K) K) ↑J = (algebraMap (𝓞 K) K) ↑ε / σ ((algebraMap (𝓞 K) K) ↑ε)
hε'✝ : ∀ (x : Fin (r + 1)), ↑↑p ∣ ι x.castSucc
this : i✝ = Fin.last (r + 1)
h : ℕ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ (h + 1))
hE :
  (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h + 1 - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
ε' : (𝓞 k)ˣ
hε' : ε' ^ ↑p = NE
hζ'' : IsPrimitiveRoot ((algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h)) ↑p
NE_p_pow : (algebraMap (𝓞 K) K) ((↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹) ^ ↑p) = (algebraMap (𝓞 K) K) 1
i : ℕ
e : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ h) ^ i = ↑(↑E * (algebraMap (𝓞 k) (𝓞 K)) ↑ε'⁻¹)
⊢ E = (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ((ζ ^ ↑p ^ h) ^ i * ε')