Theorem: solvableByRad.induction2
File: Mathlib/FieldTheory/AbelRuffini.lean

Proof steps:
state_before: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α✝ : E
α β γ : ↥(solvableByRad F E)
hγ : γ ∈ F⟮α, β⟯
hα : P α
hβ : P β
⊢ P γ
state_after: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α✝ : E
α β γ : ↥(solvableByRad F E)
hγ : γ ∈ F⟮α, β⟯
hα : P α
hβ : P β
p : F[X] := minpoly F α
⊢ P γ
tactic: let p := minpoly F α
state_before: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α✝ : E
α β γ : ↥(solvableByRad F E)
hγ : γ ∈ F⟮α, β⟯
hα : P α
hβ : P β
p : F[X] := minpoly F α
⊢ P γ
state_after: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α✝ : E
α β γ : ↥(solvableByRad F E)
hγ : γ ∈ F⟮α, β⟯
hα : P α
hβ : P β
p : F[X] := minpoly F α
q : F[X] := minpoly F β
⊢ P γ
tactic: let q := minpoly F β
state_before: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α✝ : E
α β γ : ↥(solvableByRad F E)
hγ : γ ∈ F⟮α, β⟯
hα : P α
hβ : P β
p : F[X] := minpoly F α
q : F[X] := minpoly F β
⊢ P γ
state_after: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α✝ : E
α β γ : ↥(solvableByRad F E)
hγ : γ ∈ F⟮α, β⟯
hα : P α
hβ : P β
p : F[X] := minpoly F α
q : F[X] := minpoly F β
hpq :
  Splits (algebraMap F (p * q).SplittingField) (minpoly F α) ∧
    Splits (algebraMap F (p * q).SplittingField) (minpoly F β)
⊢ P γ
tactic: have hpq := Polynomial.splits_of_splits_mul _
    (mul_ne_zero (minpoly.ne_zero (isIntegral α)) (minpoly.ne_zero (isIntegral β)))
    (SplittingField.splits (p * q))
state_before: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α✝ : E
α β γ : ↥(solvableByRad F E)
hγ : γ ∈ F⟮α, β⟯
hα : P α
hβ : P β
p : F[X] := minpoly F α
q : F[X] := minpoly F β
hpq :
  Splits (algebraMap F (p * q).SplittingField) (minpoly F α) ∧
    Splits (algebraMap F (p * q).SplittingField) (minpoly F β)
⊢ P γ
state_after: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α✝ : E
α β γ : ↥(solvableByRad F E)
hγ : γ ∈ F⟮α, β⟯
hα : P α
hβ : P β
p : F[X] := minpoly F α
q : F[X] := minpoly F β
hpq :
  Splits (algebraMap F (p * q).SplittingField) (minpoly F α) ∧
    Splits (algebraMap F (p * q).SplittingField) (minpoly F β)
f : ↥F⟮α, β⟯ →ₐ[F] (p * q).SplittingField := Classical.choice ⋯
⊢ P γ
tactic: let f : ↥F⟮α, β⟯ →ₐ[F] (p * q).SplittingField :=
    Classical.choice <| nonempty_algHom_adjoin_of_splits <| by
      intro x hx
      simp only [Set.mem_insert_iff, Set.mem_singleton_iff] at hx
      cases hx with rw [hx]
      | inl hx => exact ⟨isIntegral α, hpq.1⟩
      | inr hx => exact ⟨isIntegral β, hpq.2⟩
state_before: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α✝ : E
α β γ : ↥(solvableByRad F E)
hγ : γ ∈ F⟮α, β⟯
hα : P α
hβ : P β
p : F[X] := minpoly F α
q : F[X] := minpoly F β
hpq :
  Splits (algebraMap F (p * q).SplittingField) (minpoly F α) ∧
    Splits (algebraMap F (p * q).SplittingField) (minpoly F β)
f : ↥F⟮α, β⟯ →ₐ[F] (p * q).SplittingField := Classical.choice ⋯
⊢ P γ
state_after: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α✝ : E
α β γ : ↥(solvableByRad F E)
hγ : γ ∈ F⟮α, β⟯
hα : P α
hβ : P β
p : F[X] := minpoly F α
q : F[X] := minpoly F β
hpq :
  Splits (algebraMap F (p * q).SplittingField) (minpoly F α) ∧
    Splits (algebraMap F (p * q).SplittingField) (minpoly F β)
f : ↥F⟮α, β⟯ →ₐ[F] (p * q).SplittingField := Classical.choice ⋯
key : minpoly F γ = minpoly F (f ⟨γ, hγ⟩)
⊢ P γ
tactic: have key : minpoly F γ = minpoly F (f ⟨γ, hγ⟩) := by
    refine minpoly.eq_of_irreducible_of_monic
      (minpoly.irreducible (isIntegral γ)) ?_ (minpoly.monic (isIntegral γ))
    suffices aeval (⟨γ, hγ⟩ : F⟮α, β⟯) (minpoly F γ) = 0 by
      rw [aeval_algHom_apply, this, map_zero]
    apply (algebraMap (↥F⟮α, β⟯) (solvableByRad F E)).injective
    simp only [map_zero, _root_.map_eq_zero]
    -- Porting note: end of the proof was `exact minpoly.aeval F γ`.
    apply Subtype.val_injective
    -- This used to be `simp`, but we need `erw` and `simp` after leanprover/lean4#2644
    erw [Polynomial.aeval_subalgebra_coe (minpoly F γ)]
    simp
state_before: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α✝ : E
α β γ : ↥(solvableByRad F E)
hγ : γ ∈ F⟮α, β⟯
hα : P α
hβ : P β
p : F[X] := minpoly F α
q : F[X] := minpoly F β
hpq :
  Splits (algebraMap F (p * q).SplittingField) (minpoly F α) ∧
    Splits (algebraMap F (p * q).SplittingField) (minpoly F β)
f : ↥F⟮α, β⟯ →ₐ[F] (p * q).SplittingField := Classical.choice ⋯
key : minpoly F γ = minpoly F (f ⟨γ, hγ⟩)
⊢ P γ
state_after: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α✝ : E
α β γ : ↥(solvableByRad F E)
hγ : γ ∈ F⟮α, β⟯
hα : P α
hβ : P β
p : F[X] := minpoly F α
q : F[X] := minpoly F β
hpq :
  Splits (algebraMap F (p * q).SplittingField) (minpoly F α) ∧
    Splits (algebraMap F (p * q).SplittingField) (minpoly F β)
f : ↥F⟮α, β⟯ →ₐ[F] (p * q).SplittingField := Classical.choice ⋯
key : minpoly F γ = minpoly F (f ⟨γ, hγ⟩)
⊢ IsSolvable (minpoly F (f ⟨γ, hγ⟩)).Gal
tactic: rw [P, key]
state_before: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α✝ : E
α β γ : ↥(solvableByRad F E)
hγ : γ ∈ F⟮α, β⟯
hα : P α
hβ : P β
p : F[X] := minpoly F α
q : F[X] := minpoly F β
hpq :
  Splits (algebraMap F (p * q).SplittingField) (minpoly F α) ∧
    Splits (algebraMap F (p * q).SplittingField) (minpoly F β)
f : ↥F⟮α, β⟯ →ₐ[F] (p * q).SplittingField := Classical.choice ⋯
key : minpoly F γ = minpoly F (f ⟨γ, hγ⟩)
⊢ IsSolvable (minpoly F (f ⟨γ, hγ⟩)).Gal
state_after: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α✝ : E
α β γ : ↥(solvableByRad F E)
hγ : γ ∈ F⟮α, β⟯
hα : P α
hβ : P β
p : F[X] := minpoly F α
q : F[X] := minpoly F β
hpq :
  Splits (algebraMap F (p * q).SplittingField) (minpoly F α) ∧
    Splits (algebraMap F (p * q).SplittingField) (minpoly F β)
f : ↥F⟮α, β⟯ →ₐ[F] (p * q).SplittingField := Classical.choice ⋯
key : minpoly F γ = minpoly F (f ⟨γ, hγ⟩)
⊢ Normal F (p * q).SplittingField
tactic: refine gal_isSolvable_of_splits ⟨Normal.splits ?_ (f ⟨γ, hγ⟩)⟩ (gal_mul_isSolvable hα hβ)
state_before: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α✝ : E
α β γ : ↥(solvableByRad F E)
hγ : γ ∈ F⟮α, β⟯
hα : P α
hβ : P β
p : F[X] := minpoly F α
q : F[X] := minpoly F β
hpq :
  Splits (algebraMap F (p * q).SplittingField) (minpoly F α) ∧
    Splits (algebraMap F (p * q).SplittingField) (minpoly F β)
f : ↥F⟮α, β⟯ →ₐ[F] (p * q).SplittingField := Classical.choice ⋯
key : minpoly F γ = minpoly F (f ⟨γ, hγ⟩)
⊢ Normal F (p * q).SplittingField
state_after: no goals
tactic: apply SplittingField.instNormal