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field_theory.normal

Normal field extensions #

In this file we define normal field extensions and prove that for a finite extension, being normal is the same as being a splitting field (normal.of_is_splitting_field and normal.exists_is_splitting_field).

Main Definitions #

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@[class]
structure normal (F : Type u_1) (K : Type u_2) [field F] [field K] [algebra F K] :
Prop

Typeclass for normal field extension: K is a normal extension of F iff the minimal polynomial of every element x in K splits in K, i.e. every conjugate of x is in K.

Instances
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theorem normal.is_integral {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (h : normal F K) (x : K) :
is_integral F x
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theorem normal.splits {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (h : normal F K) (x : K) :
polynomial.splits (algebra_map F K) (minpoly F x)
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theorem normal_iff {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] :
normal F K ↔ ∀ (x : K), is_integral F x ∧ polynomial.splits (algebra_map F K) (minpoly F x)
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theorem normal.out {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] :
normal F K → ∀ (x : K), is_integral F x ∧ polynomial.splits (algebra_map F K) (minpoly F x)
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@[protected, instance]
def normal_self (F : Type u_1) [field F] :
normal F F
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theorem normal.exists_is_splitting_field (F : Type u_1) (K : Type u_2) [field F] [field K] [algebra F K] [h : normal F K] [finite_dimensional F K] :
∃ (p : F[X]), polynomial.is_splitting_field F K p
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theorem normal.tower_top_of_normal (F : Type u_1) (K : Type u_2) [field F] [field K] [algebra F K] (E : Type u_3) [field E] [algebra F E] [algebra K E] [is_scalar_tower F K E] [h : normal F E] :
normal K E
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theorem alg_hom.normal_bijective (F : Type u_1) (K : Type u_2) [field F] [field K] [algebra F K] (E : Type u_3) [field E] [algebra F E] [algebra K E] [is_scalar_tower F K E] [h : normal F E] (ϕ : E →ₐ[F] K) :
function.bijective ⇑ϕ
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theorem normal.of_alg_equiv {F : Type u_1} [field F] {E : Type u_3} [field E] [algebra F E] {E' : Type u_4} [field E'] [algebra F E'] [h : normal F E] (f : E ≃ₐ[F] E') :
normal F E'
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theorem alg_equiv.transfer_normal {F : Type u_1} [field F] {E : Type u_3} [field E] [algebra F E] {E' : Type u_4} [field E'] [algebra F E'] (f : E ≃ₐ[F] E') :
normal F E ↔ normal F E'
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theorem normal.of_is_splitting_field {F : Type u_1} [field F] {E : Type u_3} [field E] [algebra F E] (p : F[X]) [hFEp : polynomial.is_splitting_field F E p] :
normal F E
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@[protected, instance]
def polynomial.splitting_field.normal {F : Type u_1} [field F] (p : F[X]) :
normal F p.splitting_field
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def alg_hom.restrict_normal_aux {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (ϕ : K →ₐ[F] K) (E : Type u_3) [field E] [algebra F E] [algebra E K] [is_scalar_tower F E K] [h : normal F E] :
↥((is_scalar_tower.to_alg_hom F E K).range) →ₐ[F] ↥((is_scalar_tower.to_alg_hom F E K).range)

Restrict algebra homomorphism to image of normal subfield

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noncomputable def alg_hom.restrict_normal {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (ϕ : K →ₐ[F] K) (E : Type u_3) [field E] [algebra F E] [algebra E K] [is_scalar_tower F E K] [normal F E] :
E →ₐ[F] E

Restrict algebra homomorphism to normal subfield

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@[simp]
theorem alg_hom.restrict_normal_commutes {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (ϕ : K →ₐ[F] K) (E : Type u_3) [field E] [algebra F E] [algebra E K] [is_scalar_tower F E K] [normal F E] (x : E) :
⇑(algebra_map E K) (⇑(ϕ.restrict_normal E) x) = ⇑ϕ (⇑(algebra_map E K) x)
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theorem alg_hom.restrict_normal_comp {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (ϕ ψ : K →ₐ[F] K) (E : Type u_3) [field E] [algebra F E] [algebra E K] [is_scalar_tower F E K] [normal F E] :
(ϕ.restrict_normal E).comp (ψ.restrict_normal E) = (ϕ.comp ψ).restrict_normal E
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noncomputable def alg_equiv.restrict_normal {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (χ : K ≃ₐ[F] K) (E : Type u_3) [field E] [algebra F E] [algebra E K] [is_scalar_tower F E K] [h : normal F E] :
E ≃ₐ[F] E

Restrict algebra isomorphism to a normal subfield

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@[simp]
theorem alg_equiv.restrict_normal_commutes {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (χ : K ≃ₐ[F] K) (E : Type u_3) [field E] [algebra F E] [algebra E K] [is_scalar_tower F E K] [normal F E] (x : E) :
⇑(algebra_map E K) (⇑(χ.restrict_normal E) x) = ⇑χ (⇑(algebra_map E K) x)
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theorem alg_equiv.restrict_normal_trans {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (χ ω : K ≃ₐ[F] K) (E : Type u_3) [field E] [algebra F E] [algebra E K] [is_scalar_tower F E K] [normal F E] :
(χ.trans «ω»).restrict_normal E = (χ.restrict_normal E).trans («ω».restrict_normal E)
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noncomputable def alg_equiv.restrict_normal_hom {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (E : Type u_3) [field E] [algebra F E] [algebra E K] [is_scalar_tower F E K] [normal F E] :
(K ≃ₐ[F] K) →* E ≃ₐ[F] E

Restriction to an normal subfield as a group homomorphism

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noncomputable def alg_hom.lift_normal {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (ϕ : K →ₐ[F] K) (E : Type u_3) [field E] [algebra F E] [algebra K E] [is_scalar_tower F K E] [h : normal F E] :
E →ₐ[F] E

If E/K/F is a tower of fields with E/F normal then we can lift an algebra homomorphism ϕ : K →ₐ[F] K to ϕ.lift_normal E : E →ₐ[F] E.

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@[simp]
theorem alg_hom.lift_normal_commutes {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (ϕ : K →ₐ[F] K) (E : Type u_3) [field E] [algebra F E] [algebra K E] [is_scalar_tower F K E] [normal F E] (x : K) :
⇑(ϕ.lift_normal E) (⇑(algebra_map K E) x) = ⇑(algebra_map K E) (⇑ϕ x)
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@[simp]
theorem alg_hom.restrict_lift_normal {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (ϕ : K →ₐ[F] K) (E : Type u_3) [field E] [algebra F E] [algebra K E] [is_scalar_tower F K E] [normal F K] [normal F E] :
(ϕ.lift_normal E).restrict_normal K = ϕ
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noncomputable def alg_equiv.lift_normal {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (χ : K ≃ₐ[F] K) (E : Type u_3) [field E] [algebra F E] [algebra K E] [is_scalar_tower F K E] [normal F E] :
E ≃ₐ[F] E

If E/K/F is a tower of fields with E/F normal then we can lift an algebra isomorphism ϕ : K ≃ₐ[F] K to ϕ.lift_normal E : E ≃ₐ[F] E.

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@[simp]
theorem alg_equiv.lift_normal_commutes {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (χ : K ≃ₐ[F] K) (E : Type u_3) [field E] [algebra F E] [algebra K E] [is_scalar_tower F K E] [normal F E] (x : K) :
⇑(χ.lift_normal E) (⇑(algebra_map K E) x) = ⇑(algebra_map K E) (⇑χ x)
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@[simp]
theorem alg_equiv.restrict_lift_normal {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (χ : K ≃ₐ[F] K) (E : Type u_3) [field E] [algebra F E] [algebra K E] [is_scalar_tower F K E] [normal F K] [normal F E] :
(χ.lift_normal E).restrict_normal K = χ
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theorem alg_equiv.restrict_normal_hom_surjective {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (E : Type u_3) [field E] [algebra F E] [algebra K E] [is_scalar_tower F K E] [normal F K] [normal F E] :
function.surjective ⇑(alg_equiv.restrict_normal_hom K)
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theorem is_solvable_of_is_scalar_tower (F : Type u_1) (K : Type u_2) [field F] [field K] [algebra F K] (E : Type u_3) [field E] [algebra F E] [algebra K E] [is_scalar_tower F K E] [normal F K] [h1 : is_solvable (K ≃ₐ[F] K)] [h2 : is_solvable (E ≃ₐ[K] E)] :
is_solvable (E ≃ₐ[F] E)