-- Mathlib/NumberTheory/JacobiSum/Basic.lean
/-- If `χ` and `ψ` are multiplicative characters of order dividing `n` on a finite field `F`
with values in an integral domain `R` and `μ` is a primitive `n`th root of unity in `R`,
then `J(χ,ψ) = -1 + z*(μ - 1)^2` for some `z ∈ ℤ[μ] ⊆ R`. (We assume that `#F ≡ 1 mod n`.)
Note that we do not state this as a divisibility in `R`, as this would give a weaker statement. -/
lemma exists_jacobiSum_eq_neg_one_add {n : ℕ} (hn : 2 < n) {χ ψ : MulChar F R}
    {μ : R} (hχ : χ ^ n = 1) (hψ : ψ ^ n = 1) (hn' : n ∣ Fintype.card F - 1)
    (hμ : IsPrimitiveRoot μ n) :
    ∃ z ∈ Algebra.adjoin ℤ {μ}, jacobiSum χ ψ = -1 + z * (μ - 1) ^ 2

Proof state:
case neg
F : Type u_1
R : Type u_2
inst✝³ : Fintype F
inst✝² : Field F
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
hn : 2 < n
χ ψ : MulChar F R
μ : R
hχ : χ ^ n = 1
hψ : ψ ^ n = 1
hμ : IsPrimitiveRoot μ n
q : ℕ
hq : Fintype.card F = n * q + 1
z₁ : R
hz₁ : z₁ ∈ Algebra.adjoin ℤ {μ}
Hz₁ : ↑n = z₁ * (μ - 1) ^ 2
hχ₀ : ¬χ = 1
hψ₀ : ¬ψ = 1
this : NeZero n
H : ∀ (x : F), ∃ z ∈ Algebra.adjoin ℤ {μ}, (χ x - 1) * (ψ (1 - x) - 1) = z * (μ - 1) ^ 2
⊢ ∃ z ∈ Algebra.adjoin ℤ {μ},
    0 + 0 - ↑(n * q + 1) + ∑ x ∈ univ \ {0, 1}, (χ x - 1) * (ψ (1 - x) - 1) = -1 + z * (μ - 1) ^ 2

Tactic:
have Hcs x := (H x).choose_spec

Result:
case neg
F : Type u_1
R : Type u_2
inst✝³ : Fintype F
inst✝² : Field F
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
hn : 2 < n
χ ψ : MulChar F R
μ : R
hχ : χ ^ n = 1
hψ : ψ ^ n = 1
hμ : IsPrimitiveRoot μ n
q : ℕ
hq : Fintype.card F = n * q + 1
z₁ : R
hz₁ : z₁ ∈ Algebra.adjoin ℤ {μ}
Hz₁ : ↑n = z₁ * (μ - 1) ^ 2
hχ₀ : ¬χ = 1
hψ₀ : ¬ψ = 1
this : NeZero n
H : ∀ (x : F), ∃ z ∈ Algebra.adjoin ℤ {μ}, (χ x - 1) * (ψ (1 - x) - 1) = z * (μ - 1) ^ 2
Hcs : ∀ (x : F), ⋯.choose ∈ Algebra.adjoin ℤ {μ} ∧ (χ x - 1) * (ψ (1 - x) - 1) = ⋯.choose * (μ - 1) ^ 2
⊢ ∃ z ∈ Algebra.adjoin ℤ {μ},
    0 + 0 - ↑(n * q + 1) + ∑ x ∈ univ \ {0, 1}, (χ x - 1) * (ψ (1 - x) - 1) = -1 + z * (μ - 1) ^ 2

Proof state:
case intro
F : Type u_1
R : Type u_2
inst✝³ : Fintype F
inst✝² : Field F
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
hn : 2 < n
χ ψ : MulChar F R
μ : R
hχ : χ ^ n = 1
hψ : ψ ^ n = 1
hμ : IsPrimitiveRoot μ n
q : ℕ
hq : Fintype.card F = n * q + 1
⊢ ∃ z ∈ Algebra.adjoin ℤ {μ}, jacobiSum χ ψ = -1 + z * (μ - 1) ^ 2

Tactic:
obtain ⟨z₁, hz₁, Hz₁⟩ := hμ.self_sub_one_pow_dvd_order hn

Result:
case intro.intro.intro
F : Type u_1
R : Type u_2
inst✝³ : Fintype F
inst✝² : Field F
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
hn : 2 < n
χ ψ : MulChar F R
μ : R
hχ : χ ^ n = 1
hψ : ψ ^ n = 1
hμ : IsPrimitiveRoot μ n
q : ℕ
hq : Fintype.card F = n * q + 1
z₁ : R
hz₁ : z₁ ∈ Algebra.adjoin ℤ {μ}
Hz₁ : ↑n = z₁ * (μ - 1) ^ 2
⊢ ∃ z ∈ Algebra.adjoin ℤ {μ}, jacobiSum χ ψ = -1 + z * (μ - 1) ^ 2

Proof state:
case neg
F : Type u_1
R : Type u_2
inst✝³ : Fintype F
inst✝² : Field F
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
hn : 2 < n
χ ψ : MulChar F R
μ : R
hχ : χ ^ n = 1
hψ : ψ ^ n = 1
hμ : IsPrimitiveRoot μ n
q : ℕ
hq : Fintype.card F = n * q + 1
z₁ : R
hz₁ : z₁ ∈ Algebra.adjoin ℤ {μ}
Hz₁ : ↑n = z₁ * (μ - 1) ^ 2
hχ₀ : ¬χ = 1
hψ₀ : ¬ψ = 1
⊢ ∃ z ∈ Algebra.adjoin ℤ {μ},
    0 + 0 - ↑(n * q + 1) + ∑ x ∈ univ \ {0, 1}, (χ x - 1) * (ψ (1 - x) - 1) = -1 + z * (μ - 1) ^ 2

Tactic:
have : NeZero n := ⟨by omega⟩

Result:
case neg
F : Type u_1
R : Type u_2
inst✝³ : Fintype F
inst✝² : Field F
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
hn : 2 < n
χ ψ : MulChar F R
μ : R
hχ : χ ^ n = 1
hψ : ψ ^ n = 1
hμ : IsPrimitiveRoot μ n
q : ℕ
hq : Fintype.card F = n * q + 1
z₁ : R
hz₁ : z₁ ∈ Algebra.adjoin ℤ {μ}
Hz₁ : ↑n = z₁ * (μ - 1) ^ 2
hχ₀ : ¬χ = 1
hψ₀ : ¬ψ = 1
this : NeZero n
⊢ ∃ z ∈ Algebra.adjoin ℤ {μ},
    0 + 0 - ↑(n * q + 1) + ∑ x ∈ univ \ {0, 1}, (χ x - 1) * (ψ (1 - x) - 1) = -1 + z * (μ - 1) ^ 2

Proof state:
case neg.refine_2
F : Type u_1
R : Type u_2
inst✝³ : Fintype F
inst✝² : Field F
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
hn : 2 < n
χ ψ : MulChar F R
μ : R
hχ : χ ^ n = 1
hψ : ψ ^ n = 1
hμ : IsPrimitiveRoot μ n
q : ℕ
hq : Fintype.card F = n * q + 1
z₁ : R
hz₁ : z₁ ∈ Algebra.adjoin ℤ {μ}
Hz₁ : ↑n = z₁ * (μ - 1) ^ 2
hχ₀ : ¬χ = 1
hψ₀ : ¬ψ = 1
this : NeZero n
H : ∀ (x : F), ∃ z ∈ Algebra.adjoin ℤ {μ}, (χ x - 1) * (ψ (1 - x) - 1) = z * (μ - 1) ^ 2
Hcs : ∀ (x : F), ⋯.choose ∈ Algebra.adjoin ℤ {μ} ∧ (χ x - 1) * (ψ (1 - x) - 1) = ⋯.choose * (μ - 1) ^ 2
⊢ 0 + 0 - ↑(n * q + 1) + ∑ x ∈ univ \ {0, 1}, ⋯.choose * (μ - 1) ^ 2 =
    -1 + (-↑q * z₁ + ∑ x ∈ univ \ {0, 1}, ⋯.choose) * (μ - 1) ^ 2

Tactic:
rw [← Finset.sum_mul, Nat.cast_add, Nat.cast_mul, Hz₁]

Result:
case neg.refine_2
F : Type u_1
R : Type u_2
inst✝³ : Fintype F
inst✝² : Field F
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
hn : 2 < n
χ ψ : MulChar F R
μ : R
hχ : χ ^ n = 1
hψ : ψ ^ n = 1
hμ : IsPrimitiveRoot μ n
q : ℕ
hq : Fintype.card F = n * q + 1
z₁ : R
hz₁ : z₁ ∈ Algebra.adjoin ℤ {μ}
Hz₁ : ↑n = z₁ * (μ - 1) ^ 2
hχ₀ : ¬χ = 1
hψ₀ : ¬ψ = 1
this : NeZero n
H : ∀ (x : F), ∃ z ∈ Algebra.adjoin ℤ {μ}, (χ x - 1) * (ψ (1 - x) - 1) = z * (μ - 1) ^ 2
Hcs : ∀ (x : F), ⋯.choose ∈ Algebra.adjoin ℤ {μ} ∧ (χ x - 1) * (ψ (1 - x) - 1) = ⋯.choose * (μ - 1) ^ 2
⊢ 0 + 0 - (z₁ * (μ - 1) ^ 2 * ↑q + ↑1) + (∑ i ∈ univ \ {0, 1}, ⋯.choose) * (μ - 1) ^ 2 =
    -1 + (-↑q * z₁ + ∑ x ∈ univ \ {0, 1}, ⋯.choose) * (μ - 1) ^ 2

Proof state:
F : Type u_1
R : Type u_2
inst✝³ : Fintype F
inst✝² : Field F
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
hn : 2 < n
χ ψ : MulChar F R
μ : R
hχ : χ ^ n = 1
hψ : ψ ^ n = 1
hn' : n ∣ Fintype.card F - 1
hμ : IsPrimitiveRoot μ n
⊢ ∃ z ∈ Algebra.adjoin ℤ {μ}, jacobiSum χ ψ = -1 + z * (μ - 1) ^ 2

Tactic:
obtain ⟨q, hq⟩ := hn'

Result:
case intro
F : Type u_1
R : Type u_2
inst✝³ : Fintype F
inst✝² : Field F
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
hn : 2 < n
χ ψ : MulChar F R
μ : R
hχ : χ ^ n = 1
hψ : ψ ^ n = 1
hμ : IsPrimitiveRoot μ n
q : ℕ
hq : Fintype.card F - 1 = n * q
⊢ ∃ z ∈ Algebra.adjoin ℤ {μ}, jacobiSum χ ψ = -1 + z * (μ - 1) ^ 2

Proof state:
case pos
F : Type u_1
R : Type u_2
inst✝³ : Fintype F
inst✝² : Field F
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
hn : 2 < n
χ ψ : MulChar F R
μ : R
hχ : χ ^ n = 1
hψ : ψ ^ n = 1
hμ : IsPrimitiveRoot μ n
q : ℕ
hq : Fintype.card F = n * q + 1
z₁ : R
hz₁ : z₁ ∈ Algebra.adjoin ℤ {μ}
Hz₁ : ↑n = z₁ * (μ - 1) ^ 2
hχ₀ : χ = 1
hψ₀ : ψ = 1
⊢ z₁ * (μ - 1) ^ 2 * ↑q + ↑1 - 2 = -1 + ↑q * z₁ * (μ - 1) ^ 2

Tactic:
ring

Result:
no goals

Proof state:
case neg.refine_1.refine_1
F : Type u_1
R : Type u_2
inst✝³ : Fintype F
inst✝² : Field F
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
hn : 2 < n
χ ψ : MulChar F R
μ : R
hχ : χ ^ n = 1
hψ : ψ ^ n = 1
hμ : IsPrimitiveRoot μ n
q : ℕ
hq : Fintype.card F = n * q + 1
z₁ : R
hz₁ : z₁ ∈ Algebra.adjoin ℤ {μ}
Hz₁ : ↑n = z₁ * (μ - 1) ^ 2
hχ₀ : ¬χ = 1
hψ₀ : ¬ψ = 1
this : NeZero n
H : ∀ (x : F), ∃ z ∈ Algebra.adjoin ℤ {μ}, (χ x - 1) * (ψ (1 - x) - 1) = z * (μ - 1) ^ 2
Hcs : ∀ (x : F), ⋯.choose ∈ Algebra.adjoin ℤ {μ} ∧ (χ x - 1) * (ψ (1 - x) - 1) = ⋯.choose * (μ - 1) ^ 2
⊢ ↑q ∈ Algebra.adjoin ℤ {μ}

Tactic:
exact Subalgebra.natCast_mem ..

Result:
no goals