-- Mathlib/FieldTheory/AbelRuffini.lean
theorem induction2 {α β γ : solvableByRad F E} (hγ : γ ∈ F⟮α, β⟯) (hα : P α) (hβ : P β) : P γ

Proof state:
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F 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 ⋯
⊢ minpoly F γ = minpoly F (f ⟨γ, hγ⟩)

Tactic:
refine minpoly.eq_of_irreducible_of_monic
  (minpoly.irreducible (isIntegral γ)) ?_ (minpoly.monic (isIntegral γ))

Result:
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F 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 ⋯
⊢ (aeval (f ⟨γ, hγ⟩)) (minpoly F γ) = 0

Proof state:
case a.a
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F 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 ⋯
⊢ ↑((aeval ⟨γ, hγ⟩) (minpoly F γ)) = ↑0

Tactic:
erw [Polynomial.aeval_subalgebra_coe (minpoly F γ)]

Result:
case a.a
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F 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 ⋯
⊢ (aeval ↑⟨γ, hγ⟩) (minpoly F γ) = ↑0

Proof state:
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F 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 ⋯
this : (aeval ⟨γ, hγ⟩) (minpoly F γ) = 0
⊢ (aeval (f ⟨γ, hγ⟩)) (minpoly F γ) = 0

Tactic:
rw [aeval_algHom_apply, this, map_zero]

Result:
no goals

Proof state:
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F 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:
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]
  apply Subtype.val_injective
  erw [Polynomial.aeval_subalgebra_coe (minpoly F γ)]
  simp

Result:
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F 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

Proof state:
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F 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:
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⟩

Result:
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F 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

Proof state:
case a
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F 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 ⋯
⊢ (algebraMap ↥F⟮α, β⟯ ↥(solvableByRad F E)) ((aeval ⟨γ, hγ⟩) (minpoly F γ)) =
    (algebraMap ↥F⟮α, β⟯ ↥(solvableByRad F E)) 0

Tactic:
simp only [map_zero, _root_.map_eq_zero]

Result:
case a
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F 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 ⋯
⊢ (aeval ⟨γ, hγ⟩) (minpoly F γ) = 0

Proof state:
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F 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:
apply SplittingField.instNormal

Result:
no goals

Proof state:
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F 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:
refine gal_isSolvable_of_splits ⟨Normal.splits ?_ (f ⟨γ, hγ⟩)⟩ (gal_mul_isSolvable hα hβ)

Result:
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F 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