internlm/Lean-Github huggingface dataset:
Theorem: KummersLemma.separable_poly
Proof state before:
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝ : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
⊢ (map (Ideal.Quotient.mk I) (poly hp hζ u hcong)).Separable
Tactic:
letI : Module (𝓞 K) (𝓞 L) := Algebra.toModule
Proof state after:
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝¹ : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝ : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this : Module (𝓞 K) (𝓞 L) := Algebra.toModule
⊢ (map (Ideal.Quotient.mk I) (poly hp hζ u hcong)).Separable
Theorem: KummersLemma.separable_poly
Proof state before:
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝¹ : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝ : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this : Module (𝓞 K) (𝓞 L) := Algebra.toModule
⊢ (map (Ideal.Quotient.mk I) (poly hp hζ u hcong)).Separable
Tactic:
letI := Ideal.Quotient.commRing J
Proof state after:
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝² : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝¹ : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
⊢ (map (Ideal.Quotient.mk I) (poly hp hζ u hcong)).Separable
Theorem: KummersLemma.separable_poly
Proof state before:
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝² : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝¹ : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
⊢ (map (Ideal.Quotient.mk I) (poly hp hζ u hcong)).Separable
Tactic:
let i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap _
  (algebraMap (𝓞 K) (𝓞 L)) Ideal.le_comap_map
Proof state after:
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝² : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝¹ : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap J (algebraMap (𝓞 K) (𝓞 L)) ⋯
⊢ (map (Ideal.Quotient.mk I) (poly hp hζ u hcong)).Separable
Theorem: KummersLemma.separable_poly
Proof state before:
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝³ : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝² : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝¹ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this✝ : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap J (algebraMap (𝓞 K) (𝓞 L)) ⋯
this : Nontrivial (𝓞 L ⧸ J)
⊢ (map (Ideal.Quotient.mk I) (poly hp hζ u hcong)).Separable
Tactic:
rw [← Polynomial.separable_map' i, map_map, Ideal.quotientMap_comp_mk, ← map_map]
Proof state after:
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝³ : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝² : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝¹ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this✝ : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap J (algebraMap (𝓞 K) (𝓞 L)) ⋯
this : Nontrivial (𝓞 L ⧸ J)
⊢ (map (Ideal.Quotient.mk J) (map (algebraMap (𝓞 K) (𝓞 L)) (poly hp hζ u hcong))).Separable
Theorem: KummersLemma.separable_poly
Proof state before:
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝³ : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝² : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝¹ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this✝ : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap J (algebraMap (𝓞 K) (𝓞 L)) ⋯
this : Nontrivial (𝓞 L ⧸ J)
⊢ (map (Ideal.Quotient.mk J) (map (algebraMap (𝓞 K) (𝓞 L)) (poly hp hζ u hcong))).Separable
Tactic:
apply Separable.map
Proof state after:
case h
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝³ : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝² : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝¹ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this✝ : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap J (algebraMap (𝓞 K) (𝓞 L)) ⋯
this : Nontrivial (𝓞 L ⧸ J)
⊢ (map (algebraMap (𝓞 K) (𝓞 L)) (poly hp hζ u hcong)).Separable
Theorem: KummersLemma.separable_poly
Proof state before:
case h
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝³ : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝² : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝¹ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this✝ : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap J (algebraMap (𝓞 K) (𝓞 L)) ⋯
this : Nontrivial (𝓞 L ⧸ J)
⊢ (map (algebraMap (𝓞 K) (𝓞 L)) (poly hp hζ u hcong)).Separable
Tactic:
apply separable_poly_aux hp hζ u hcong
Proof state after:
case h.e
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝³ : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝² : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝¹ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this✝ : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap J (algebraMap (𝓞 K) (𝓞 L)) ⋯
this : Nontrivial (𝓞 L ⧸ J)
⊢ ?h.α ^ ↑p = (algebraMap K L) ((algebraMap (𝓞 K) K) ↑u)

case h.α
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝³ : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝² : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝¹ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this✝ : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap J (algebraMap (𝓞 K) (𝓞 L)) ⋯
this : Nontrivial (𝓞 L ⧸ J)
⊢ L
Theorem: KummersLemma.separable_poly
Proof state before:
case h.e
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝³ : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝² : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝¹ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this✝ : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap J (algebraMap (𝓞 K) (𝓞 L)) ⋯
this : Nontrivial (𝓞 L ⧸ J)
⊢ ?h.α ^ ↑p = (algebraMap K L) ((algebraMap (𝓞 K) K) ↑u)

case h.α
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝³ : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝² : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝¹ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this✝ : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap J (algebraMap (𝓞 K) (𝓞 L)) ⋯
this : Nontrivial (𝓞 L ⧸ J)
⊢ L
Tactic:
exact root_X_pow_sub_C_pow _ _
Proof state after:
no goals
Theorem: KummersLemma.separable_poly
Proof state before:
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝² : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝¹ : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap J (algebraMap (𝓞 K) (𝓞 L)) ⋯
⊢ Nontrivial (𝓞 L ⧸ J)
Tactic:
apply Ideal.Quotient.nontrivial
Proof state after:
case hI
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝² : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝¹ : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap J (algebraMap (𝓞 K) (𝓞 L)) ⋯
⊢ J ≠ ⊤
Theorem: KummersLemma.separable_poly
Proof state before:
case hI.hf₁
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝² : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝¹ : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap J (algebraMap (𝓞 K) (𝓞 L)) ⋯
⊢ Function.Injective ⇑(algebraMap (𝓞 K) (𝓞 L))
Tactic:
intros x y e
Proof state after:
case hI.hf₁
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝² : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝¹ : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap J (algebraMap (𝓞 K) (𝓞 L)) ⋯
x y : 𝓞 K
e : (algebraMap (𝓞 K) (𝓞 L)) x = (algebraMap (𝓞 K) (𝓞 L)) y
⊢ x = y
Theorem: KummersLemma.separable_poly
Proof state before:
case hI.hf₁
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝² : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝¹ : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap J (algebraMap (𝓞 K) (𝓞 L)) ⋯
x y : 𝓞 K
e : (algebraMap (𝓞 K) (𝓞 L)) x = (algebraMap (𝓞 K) (𝓞 L)) y
⊢ x = y
Tactic:
ext
Proof state after:
case hI.hf₁.h
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝² : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝¹ : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap J (algebraMap (𝓞 K) (𝓞 L)) ⋯
x y : 𝓞 K
e : (algebraMap (𝓞 K) (𝓞 L)) x = (algebraMap (𝓞 K) (𝓞 L)) y
⊢ ↑x = ↑y
Theorem: KummersLemma.separable_poly
Proof state before:
case hI.hf₁.h
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝² : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝¹ : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap J (algebraMap (𝓞 K) (𝓞 L)) ⋯
x y : 𝓞 K
e : (algebraMap (𝓞 K) (𝓞 L)) x = (algebraMap (𝓞 K) (𝓞 L)) y
⊢ ↑x = ↑y
Tactic:
exact (algebraMap K L).injective (congr_arg Subtype.val e)
Proof state after:
no goals
Theorem: KummersLemma.separable_poly
Proof state before:
case hI.hf₂
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝² : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝¹ : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap J (algebraMap (𝓞 K) (𝓞 L)) ⋯
⊢ (algebraMap (𝓞 K) (𝓞 L)).IsIntegral
Tactic:
intros x
Proof state after:
case hI.hf₂
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝² : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝¹ : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap J (algebraMap (𝓞 K) (𝓞 L)) ⋯
x : 𝓞 L
⊢ (algebraMap (𝓞 K) (𝓞 L)).IsIntegralElem x
Theorem: KummersLemma.separable_poly
Proof state before:
case hI.hf₂
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝³ : Field K
inst✝² : NumberField K
inst✝¹ : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
I : Ideal (𝓞 K)
inst✝ : I.IsMaximal
L : Type u_1 := K[↑p√(algebraMap (𝓞 K) K) ↑u]
this✝² : Fact (Irreducible (X ^ ↑p - C ((algebraMap (𝓞 K) K) ↑u))) :=
  { out := X_pow_sub_C_irreducible_of_prime Fact.out hu }
J : Ideal (𝓞 L) := Ideal.map (algebraMap (𝓞 K) (𝓞 L)) I
this✝¹ : AddCommGroup (𝓞 L) := AddCommGroupWithOne.toAddCommGroup
this✝ : Module (𝓞 K) (𝓞 L) := Algebra.toModule
this : CommRing (𝓞 L ⧸ J) := Ideal.Quotient.commRing J
i : 𝓞 K ⧸ I →+* 𝓞 L ⧸ J := Ideal.quotientMap J (algebraMap (𝓞 K) (𝓞 L)) ⋯
x : 𝓞 L
⊢ (algebraMap (𝓞 K) (𝓞 L)).IsIntegralElem x
Tactic:
exact IsIntegral.tower_top (IsIntegralClosure.isIntegral ℤ L x)
Proof state after:
no goals
Theorem: KummersLemma.polyRoot_spec
Proof state before:
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
L : Type u_2
inst✝¹ : Field L
inst✝ : Algebra K L
α : L
e : α ^ ↑p = (algebraMap K L) ((algebraMap (𝓞 K) K) ↑u)
i : ℕ
⊢ α = (ζ ^ i)⁻¹ • (1 - (ζ - 1) • ↑(polyRoot hp hζ u hcong α e i))
Tactic:
apply smul_right_injective (M := L) (c := ζ ^ i) (pow_ne_zero _ <| hζ.ne_zero p.pos.ne.symm)
Proof state after:
case a
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
L : Type u_2
inst✝¹ : Field L
inst✝ : Algebra K L
α : L
e : α ^ ↑p = (algebraMap K L) ((algebraMap (𝓞 K) K) ↑u)
i : ℕ
⊢ (fun x => ζ ^ i • x) α = (fun x => ζ ^ i • x) ((ζ ^ i)⁻¹ • (1 - (ζ - 1) • ↑(polyRoot hp hζ u hcong α e i)))
Theorem: KummersLemma.polyRoot_spec
Proof state before:
case a
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : IsCyclotomicExtension {p} ℚ K
hp : p ≠ 2
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
u : (𝓞 K)ˣ
hcong : (↑hζ.unit' - 1) ^ ↑p ∣ ↑u - 1
hu : ∀ (v : K), v ^ ↑p ≠ (algebraMap (𝓞 K) K) ↑u
L : Type u_2
inst✝¹ : Field L
inst✝ : Algebra K L
α : L
e : α ^ ↑p = (algebraMap K L) ((algebraMap (𝓞 K) K) ↑u)
i : ℕ
⊢ (fun x => ζ ^ i • x) α = (fun x => ζ ^ i • x) ((ζ ^ i)⁻¹ • (1 - (ζ - 1) • ↑(polyRoot hp hζ u hcong α e i)))
Tactic:
simp only [polyRoot, map_sub, map_one, NumberField.RingOfIntegers.map_mk,
  Algebra.smul_def (ζ - 1), ← mul_div_assoc,
  mul_div_cancel_left₀ _
    ((hζ.map_of_injective (algebraMap K L).injective).sub_one_ne_zero hpri.out.one_lt),
  sub_sub_cancel, smul_smul, inv_mul_cancel (pow_ne_zero _ <| hζ.ne_zero p.pos.ne.symm), one_smul]
Proof state after:
no goals