Theorem: IntermediateField.sepDegree_adjoin_eq_of_isAlgebraic_of_isPurelyInseparable
File: Mathlib/FieldTheory/PurelyInseparable.lean

tactic: set M := adjoin F S
state_before: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : Algebra.IsAlgebraic F ↥(adjoin F S)
inst✝ : IsPurelyInseparable F E
⊢ sepDegree E ↥(adjoin E S) = sepDegree F ↥(adjoin F S)
state_after: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
⊢ sepDegree E ↥(adjoin E S) = sepDegree F ↥M

tactic: set L := adjoin E S
state_before: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
⊢ sepDegree E ↥(adjoin E S) = sepDegree F ↥M
state_after: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
⊢ sepDegree E ↥L = sepDegree F ↥M

tactic: let E' := (IsScalarTower.toAlgHom F E K).fieldRange
state_before: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
⊢ sepDegree E ↥L = sepDegree F ↥M
state_after: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
⊢ sepDegree E ↥L = sepDegree F ↥M

tactic: let j : E ≃ₐ[F] E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
state_before: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
⊢ sepDegree E ↥L = sepDegree F ↥M
state_after: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
⊢ sepDegree E ↥L = sepDegree F ↥M

tactic: have hi : M ≤ L.restrictScalars F := by
  rw [restrictScalars_adjoin_of_algEquiv (E := K) j rfl, restrictScalars_adjoin]
  exact adjoin.mono _ _ _ Set.subset_union_right
state_before: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
⊢ sepDegree E ↥L = sepDegree F ↥M
state_after: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
⊢ sepDegree E ↥L = sepDegree F ↥M

tactic: let i : M →+* L := Subsemiring.inclusion hi
state_before: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
⊢ sepDegree E ↥L = sepDegree F ↥M
state_after: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
⊢ sepDegree E ↥L = sepDegree F ↥M

tactic: letI : Algebra M L := i.toAlgebra
state_before: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
⊢ sepDegree E ↥L = sepDegree F ↥M
state_after: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this : Algebra ↥M ↥L := i.toAlgebra
⊢ sepDegree E ↥L = sepDegree F ↥M

tactic: letI : SMul M L := Algebra.toSMul
state_before: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this : Algebra ↥M ↥L := i.toAlgebra
⊢ sepDegree E ↥L = sepDegree F ↥M
state_after: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝ : Algebra ↥M ↥L := i.toAlgebra
this : SMul ↥M ↥L := Algebra.toSMul
⊢ sepDegree E ↥L = sepDegree F ↥M

tactic: haveI : IsScalarTower F M L := IsScalarTower.of_algebraMap_eq (congrFun rfl)
state_before: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝ : Algebra ↥M ↥L := i.toAlgebra
this : SMul ↥M ↥L := Algebra.toSMul
⊢ sepDegree E ↥L = sepDegree F ↥M
state_after: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝¹ : Algebra ↥M ↥L := i.toAlgebra
this✝ : SMul ↥M ↥L := Algebra.toSMul
this : IsScalarTower F ↥M ↥L
⊢ sepDegree E ↥L = sepDegree F ↥M

tactic: haveI : IsPurelyInseparable M L := by
  change IsPurelyInseparable M (extendScalars hi)
  obtain ⟨q, _⟩ := ExpChar.exists F
  have : extendScalars hi = adjoin M (E' : Set K) := restrictScalars_injective F <| by
    conv_lhs => rw [extendScalars_restrictScalars, restrictScalars_adjoin_of_algEquiv
      (E := K) j rfl, ← adjoin_self F E', adjoin_adjoin_comm]
  rw [this, isPurelyInseparable_adjoin_iff_pow_mem _ _ q]
  rintro x ⟨y, hy⟩
  obtain ⟨n, z, hz⟩ := IsPurelyInseparable.pow_mem F q y
  refine ⟨n, algebraMap F M z, ?_⟩
  rw [← IsScalarTower.algebraMap_apply, IsScalarTower.algebraMap_apply F E K, hz, ← hy, map_pow,
    AlgHom.toRingHom_eq_coe, IsScalarTower.coe_toAlgHom]
state_before: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝¹ : Algebra ↥M ↥L := i.toAlgebra
this✝ : SMul ↥M ↥L := Algebra.toSMul
this : IsScalarTower F ↥M ↥L
⊢ sepDegree E ↥L = sepDegree F ↥M
state_after: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝² : Algebra ↥M ↥L := i.toAlgebra
this✝¹ : SMul ↥M ↥L := Algebra.toSMul
this✝ : IsScalarTower F ↥M ↥L
this : IsPurelyInseparable ↥M ↥L
⊢ sepDegree E ↥L = sepDegree F ↥M

tactic: have h := lift_sepDegree_mul_lift_sepDegree_of_isAlgebraic F E L
state_before: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝² : Algebra ↥M ↥L := i.toAlgebra
this✝¹ : SMul ↥M ↥L := Algebra.toSMul
this✝ : IsScalarTower F ↥M ↥L
this : IsPurelyInseparable ↥M ↥L
⊢ sepDegree E ↥L = sepDegree F ↥M
state_after: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝² : Algebra ↥M ↥L := i.toAlgebra
this✝¹ : SMul ↥M ↥L := Algebra.toSMul
this✝ : IsScalarTower F ↥M ↥L
this : IsPurelyInseparable ↥M ↥L
h : Cardinal.lift.{w, v} (sepDegree F E) * Cardinal.lift.{v, w} (sepDegree E ↥L) = Cardinal.lift.{v, w} (sepDegree F ↥L)
⊢ sepDegree E ↥L = sepDegree F ↥M

tactic: rw [IsPurelyInseparable.sepDegree_eq_one F E, Cardinal.lift_one, one_mul] at h
state_before: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝² : Algebra ↥M ↥L := i.toAlgebra
this✝¹ : SMul ↥M ↥L := Algebra.toSMul
this✝ : IsScalarTower F ↥M ↥L
this : IsPurelyInseparable ↥M ↥L
h : Cardinal.lift.{w, v} (sepDegree F E) * Cardinal.lift.{v, w} (sepDegree E ↥L) = Cardinal.lift.{v, w} (sepDegree F ↥L)
⊢ sepDegree E ↥L = sepDegree F ↥M
state_after: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝² : Algebra ↥M ↥L := i.toAlgebra
this✝¹ : SMul ↥M ↥L := Algebra.toSMul
this✝ : IsScalarTower F ↥M ↥L
this : IsPurelyInseparable ↥M ↥L
h : Cardinal.lift.{v, w} (sepDegree E ↥L) = Cardinal.lift.{v, w} (sepDegree F ↥L)
⊢ sepDegree E ↥L = sepDegree F ↥M

tactic: rw [Cardinal.lift_injective h, ← sepDegree_mul_sepDegree_of_isAlgebraic F M L,
  IsPurelyInseparable.sepDegree_eq_one M L, mul_one]
state_before: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝² : Algebra ↥M ↥L := i.toAlgebra
this✝¹ : SMul ↥M ↥L := Algebra.toSMul
this✝ : IsScalarTower F ↥M ↥L
this : IsPurelyInseparable ↥M ↥L
h : Cardinal.lift.{v, w} (sepDegree E ↥L) = Cardinal.lift.{v, w} (sepDegree F ↥L)
⊢ sepDegree E ↥L = sepDegree F ↥M
state_after: no goals

tactic: rw [restrictScalars_adjoin_of_algEquiv (E := K) j rfl, restrictScalars_adjoin]
state_before: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
⊢ M ≤ restrictScalars F L
state_after: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
⊢ M ≤ adjoin F (↑E' ∪ S)

tactic: exact adjoin.mono _ _ _ Set.subset_union_right
state_before: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
⊢ M ≤ adjoin F (↑E' ∪ S)
state_after: no goals

tactic: change IsPurelyInseparable M (extendScalars hi)
state_before: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝¹ : Algebra ↥M ↥L := i.toAlgebra
this✝ : SMul ↥M ↥L := Algebra.toSMul
this : IsScalarTower F ↥M ↥L
⊢ IsPurelyInseparable ↥M ↥L
state_after: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝¹ : Algebra ↥M ↥L := i.toAlgebra
this✝ : SMul ↥M ↥L := Algebra.toSMul
this : IsScalarTower F ↥M ↥L
⊢ IsPurelyInseparable ↥M ↥(extendScalars hi)

tactic: obtain ⟨q, _⟩ := ExpChar.exists F
state_before: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝¹ : Algebra ↥M ↥L := i.toAlgebra
this✝ : SMul ↥M ↥L := Algebra.toSMul
this : IsScalarTower F ↥M ↥L
⊢ IsPurelyInseparable ↥M ↥(extendScalars hi)
state_after: case intro
F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝¹ : Algebra ↥M ↥L := i.toAlgebra
this✝ : SMul ↥M ↥L := Algebra.toSMul
this : IsScalarTower F ↥M ↥L
q : ℕ
h✝ : ExpChar F q
⊢ IsPurelyInseparable ↥M ↥(extendScalars hi)

tactic: have : extendScalars hi = adjoin M (E' : Set K) := restrictScalars_injective F <| by
  conv_lhs => rw [extendScalars_restrictScalars, restrictScalars_adjoin_of_algEquiv
    (E := K) j rfl, ← adjoin_self F E', adjoin_adjoin_comm]
state_before: case intro
F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝¹ : Algebra ↥M ↥L := i.toAlgebra
this✝ : SMul ↥M ↥L := Algebra.toSMul
this : IsScalarTower F ↥M ↥L
q : ℕ
h✝ : ExpChar F q
⊢ IsPurelyInseparable ↥M ↥(extendScalars hi)
state_after: case intro
F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝² : Algebra ↥M ↥L := i.toAlgebra
this✝¹ : SMul ↥M ↥L := Algebra.toSMul
this✝ : IsScalarTower F ↥M ↥L
q : ℕ
h✝ : ExpChar F q
this : extendScalars hi = adjoin ↥M ↑E'
⊢ IsPurelyInseparable ↥M ↥(extendScalars hi)

tactic: rw [this, isPurelyInseparable_adjoin_iff_pow_mem _ _ q]
state_before: case intro
F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝² : Algebra ↥M ↥L := i.toAlgebra
this✝¹ : SMul ↥M ↥L := Algebra.toSMul
this✝ : IsScalarTower F ↥M ↥L
q : ℕ
h✝ : ExpChar F q
this : extendScalars hi = adjoin ↥M ↑E'
⊢ IsPurelyInseparable ↥M ↥(extendScalars hi)
state_after: case intro
F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝² : Algebra ↥M ↥L := i.toAlgebra
this✝¹ : SMul ↥M ↥L := Algebra.toSMul
this✝ : IsScalarTower F ↥M ↥L
q : ℕ
h✝ : ExpChar F q
this : extendScalars hi = adjoin ↥M ↑E'
⊢ ∀ x ∈ ↑E', ∃ n, x ^ q ^ n ∈ (algebraMap (↥M) K).range

tactic: rintro x ⟨y, hy⟩
state_before: case intro
F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝² : Algebra ↥M ↥L := i.toAlgebra
this✝¹ : SMul ↥M ↥L := Algebra.toSMul
this✝ : IsScalarTower F ↥M ↥L
q : ℕ
h✝ : ExpChar F q
this : extendScalars hi = adjoin ↥M ↑E'
⊢ ∀ x ∈ ↑E', ∃ n, x ^ q ^ n ∈ (algebraMap (↥M) K).range
state_after: case intro.intro
F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝² : Algebra ↥M ↥L := i.toAlgebra
this✝¹ : SMul ↥M ↥L := Algebra.toSMul
this✝ : IsScalarTower F ↥M ↥L
q : ℕ
h✝ : ExpChar F q
this : extendScalars hi = adjoin ↥M ↑E'
x : K
y : E
hy : (IsScalarTower.toAlgHom F E K).toRingHom y = x
⊢ ∃ n, x ^ q ^ n ∈ (algebraMap (↥M) K).range

tactic: obtain ⟨n, z, hz⟩ := IsPurelyInseparable.pow_mem F q y
state_before: case intro.intro
F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝² : Algebra ↥M ↥L := i.toAlgebra
this✝¹ : SMul ↥M ↥L := Algebra.toSMul
this✝ : IsScalarTower F ↥M ↥L
q : ℕ
h✝ : ExpChar F q
this : extendScalars hi = adjoin ↥M ↑E'
x : K
y : E
hy : (IsScalarTower.toAlgHom F E K).toRingHom y = x
⊢ ∃ n, x ^ q ^ n ∈ (algebraMap (↥M) K).range
state_after: case intro.intro.intro.intro
F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝² : Algebra ↥M ↥L := i.toAlgebra
this✝¹ : SMul ↥M ↥L := Algebra.toSMul
this✝ : IsScalarTower F ↥M ↥L
q : ℕ
h✝ : ExpChar F q
this : extendScalars hi = adjoin ↥M ↑E'
x : K
y : E
hy : (IsScalarTower.toAlgHom F E K).toRingHom y = x
n : ℕ
z : F
hz : (algebraMap F E) z = y ^ q ^ n
⊢ ∃ n, x ^ q ^ n ∈ (algebraMap (↥M) K).range

tactic: refine ⟨n, algebraMap F M z, ?_⟩
state_before: case intro.intro.intro.intro
F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝² : Algebra ↥M ↥L := i.toAlgebra
this✝¹ : SMul ↥M ↥L := Algebra.toSMul
this✝ : IsScalarTower F ↥M ↥L
q : ℕ
h✝ : ExpChar F q
this : extendScalars hi = adjoin ↥M ↑E'
x : K
y : E
hy : (IsScalarTower.toAlgHom F E K).toRingHom y = x
n : ℕ
z : F
hz : (algebraMap F E) z = y ^ q ^ n
⊢ ∃ n, x ^ q ^ n ∈ (algebraMap (↥M) K).range
state_after: case intro.intro.intro.intro
F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝² : Algebra ↥M ↥L := i.toAlgebra
this✝¹ : SMul ↥M ↥L := Algebra.toSMul
this✝ : IsScalarTower F ↥M ↥L
q : ℕ
h✝ : ExpChar F q
this : extendScalars hi = adjoin ↥M ↑E'
x : K
y : E
hy : (IsScalarTower.toAlgHom F E K).toRingHom y = x
n : ℕ
z : F
hz : (algebraMap F E) z = y ^ q ^ n
⊢ (algebraMap (↥M) K) ((algebraMap F ↥M) z) = x ^ q ^ n

tactic: rw [← IsScalarTower.algebraMap_apply, IsScalarTower.algebraMap_apply F E K, hz, ← hy, map_pow,
  AlgHom.toRingHom_eq_coe, IsScalarTower.coe_toAlgHom]
state_before: case intro.intro.intro.intro
F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝² : Algebra ↥M ↥L := i.toAlgebra
this✝¹ : SMul ↥M ↥L := Algebra.toSMul
this✝ : IsScalarTower F ↥M ↥L
q : ℕ
h✝ : ExpChar F q
this : extendScalars hi = adjoin ↥M ↑E'
x : K
y : E
hy : (IsScalarTower.toAlgHom F E K).toRingHom y = x
n : ℕ
z : F
hz : (algebraMap F E) z = y ^ q ^ n
⊢ (algebraMap (↥M) K) ((algebraMap F ↥M) z) = x ^ q ^ n
state_after: no goals

tactic: conv_lhs => rw [extendScalars_restrictScalars, restrictScalars_adjoin_of_algEquiv
  (E := K) j rfl, ← adjoin_self F E', adjoin_adjoin_comm]
state_before: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝¹ : Algebra ↥M ↥L := i.toAlgebra
this✝ : SMul ↥M ↥L := Algebra.toSMul
this : IsScalarTower F ↥M ↥L
q : ℕ
h✝ : ExpChar F q
⊢ restrictScalars F (extendScalars hi) = restrictScalars F (adjoin ↥M ↑E')
state_after: no goals