Normal field extensions

In this file we prove that for a finite extension, being normal is the same as being a splitting field (Normal.of_isSplittingField and Normal.exists_isSplittingField).

Additional Results

• Algebra.IsQuadraticExtension.normal: the instance that a quadratic extension, given as a class Algebra.IsQuadraticExtension, is normal.

theorem Normal.exists_isSplittingField (F : Type u_1) (K : Type u_2) [Field F] [Field K] [Algebra F K] [h : Normal F K] [FiniteDimensional F K] : ∃ (p : Polynomial F), Polynomial.IsSplittingField F K p

theorem Normal.of_isSplittingField {F : Type u_1} [Field F] {E : Type u_3} [Field E] [Algebra F E] (p : Polynomial F) [hFEp : Polynomial.IsSplittingField F E p] : Normal F E

Stacks Tag 09HU (Normal part)

instance Polynomial.SplittingField.instNormal {F : Type u_1} [Field F] (p : Polynomial F) : Normal F p.SplittingField

instance IntermediateField.normal_iSup (F : Type u_1) (K : Type u_2) [Field F] [Field K] [Algebra F K] {ι : Type u_3} (t : ι → IntermediateField F K) [h : ∀ (i : ι), Normal F ↥(t i)] : Normal F ↥(⨆ (i : ι), t i)

A compositum of normal extensions is normal.

theorem IntermediateField.splits_of_mem_adjoin (F : Type u_1) (K : Type u_2) [Field F] [Field K] [Algebra F K] {L : Type u_3} [Field L] [Algebra F L] {S : Set K} (splits : ∀ x ∈ S, IsIntegral F x ∧ (Polynomial.map (algebraMap F L) (minpoly F x)).Splits) {x : K} (hx : x ∈ adjoin F S) : (Polynomial.map (algebraMap F L) (minpoly F x)).Splits

If a set of algebraic elements in a field extension K/F have minimal polynomials that split in another extension L/F, then all minimal polynomials in the intermediate field generated by the set also split in L/F.

instance IntermediateField.normal_sup (F : Type u_1) (K : Type u_2) [Field F] [Field K] [Algebra F K] (E E' : IntermediateField F K) [Normal F ↥E] [Normal F ↥E'] : Normal F ↥(E ⊔ E')

instance IntermediateField.normal_iInf (F : Type u_1) (K : Type u_2) [Field F] [Field K] [Algebra F K] {ι : Type u_3} [hι : Nonempty ι] (t : ι → IntermediateField F K) [h : ∀ (i : ι), Normal F ↥(t i)] : Normal F ↥(⨅ (i : ι), t i)

An intersection of normal extensions is normal.

instance IntermediateField.normal_inf (F : Type u_1) (K : Type u_2) [Field F] [Field K] [Algebra F K] (E E' : IntermediateField F K) [Normal F ↥E] [Normal F ↥E'] : Normal F ↥(E ⊓ E')

Stacks Tag 09HP

theorem AlgHom.fieldRange_of_normal {F : Type u_1} {K : Type u_2} [Field F] [Field K] [Algebra F K] {E : IntermediateField F K} [Normal F ↥E] (f : ↥E →ₐ[F] K) : f.fieldRange = E

noncomputable def AlgHom.liftNormal {F : Type u_1} [Field F] {K₁ : Type u_3} {K₂ : Type u_4} [Field K₁] [Field K₂] [Algebra F K₁] [Algebra F K₂] (ϕ : K₁ →ₐ[F] K₂) (E : Type u_6) [Field E] [Algebra F E] [Algebra K₁ E] [Algebra K₂ E] [IsScalarTower F K₁ E] [IsScalarTower F K₂ E] [h : Normal F E] : E →ₐ[F] E

If E/Kᵢ/F are towers of fields with E/F normal then we can lift an algebra homomorphism ϕ : K₁ →ₐ[F] K₂ to ϕ.liftNormal E : E →ₐ[F] E.

@[simp]

theorem AlgHom.liftNormal_commutes {F : Type u_1} [Field F] {K₁ : Type u_3} {K₂ : Type u_4} [Field K₁] [Field K₂] [Algebra F K₁] [Algebra F K₂] (ϕ : K₁ →ₐ[F] K₂) (E : Type u_6) [Field E] [Algebra F E] [Algebra K₁ E] [Algebra K₂ E] [IsScalarTower F K₁ E] [IsScalarTower F K₂ E] [Normal F E] (x : K₁) : (ϕ.liftNormal E) ((algebraMap K₁ E) x) = (algebraMap K₂ E) (ϕ x)

@[simp]

theorem AlgHom.restrict_liftNormal {F : Type u_1} [Field F] {K₁ : Type u_3} [Field K₁] [Algebra F K₁] (E : Type u_6) [Field E] [Algebra F E] [Algebra K₁ E] [IsScalarTower F K₁ E] (ϕ : K₁ →ₐ[F] K₁) [Normal F K₁] [Normal F E] : (ϕ.liftNormal E).restrictNormal K₁ = ϕ

noncomputable def AlgEquiv.liftNormal {F : Type u_1} [Field F] {K₁ : Type u_3} {K₂ : Type u_4} [Field K₁] [Field K₂] [Algebra F K₁] [Algebra F K₂] (χ : K₁ ≃ₐ[F] K₂) (E : Type u_6) [Field E] [Algebra F E] [Algebra K₁ E] [Algebra K₂ E] [IsScalarTower F K₁ E] [IsScalarTower F K₂ E] [Normal F E] : Gal(E/F)

If E/Kᵢ/F are towers of fields with E/F normal then we can lift an algebra isomorphism ϕ : K₁ ≃ₐ[F] K₂ to ϕ.liftNormal E : Gal(E/F).

@[simp]

theorem AlgEquiv.liftNormal_commutes {F : Type u_1} [Field F] {K₁ : Type u_3} {K₂ : Type u_4} [Field K₁] [Field K₂] [Algebra F K₁] [Algebra F K₂] (χ : K₁ ≃ₐ[F] K₂) (E : Type u_6) [Field E] [Algebra F E] [Algebra K₁ E] [Algebra K₂ E] [IsScalarTower F K₁ E] [IsScalarTower F K₂ E] [Normal F E] (x : K₁) : (χ.liftNormal E) ((algebraMap K₁ E) x) = (algebraMap K₂ E) (χ x)

@[simp]

theorem AlgEquiv.restrict_liftNormal {F : Type u_1} [Field F] {K₁ : Type u_3} [Field K₁] [Algebra F K₁] (E : Type u_6) [Field E] [Algebra F E] [Algebra K₁ E] [IsScalarTower F K₁ E] (χ : Gal(K₁/F)) [Normal F K₁] [Normal F E] : (χ.liftNormal E).restrictNormal K₁ = χ

theorem AlgEquiv.restrictNormalHom_surjective {F : Type u_1} [Field F] {K₁ : Type u_3} [Field K₁] [Algebra F K₁] (E : Type u_6) [Field E] [Algebra F E] [Algebra K₁ E] [IsScalarTower F K₁ E] [Normal F K₁] [Normal F E] : Function.Surjective ⇑(restrictNormalHom K₁)

The group homomorphism given by restricting an algebra isomorphism to a normal subfield is surjective.

theorem Normal.minpoly_eq_iff_mem_orbit {F : Type u_1} [Field F] (E : Type u_6) [Field E] [Algebra F E] [h : Normal F E] {x y : E} : minpoly F x = minpoly F y ↔ x ∈ MulAction.orbit Gal(E/F) y

theorem isSolvable_of_isScalarTower (F : Type u_1) [Field F] (K₁ : Type u_3) [Field K₁] [Algebra F K₁] (E : Type u_6) [Field E] [Algebra F E] [Algebra K₁ E] [IsScalarTower F K₁ E] [Normal F K₁] [h1 : IsSolvable Gal(K₁/F)] [h2 : IsSolvable Gal(E/K₁)] : IsSolvable Gal(E/F)

theorem minpoly.exists_algEquiv_of_root {K : Type u_6} {L : Type u_7} [Field K] [Field L] [Algebra K L] [Normal K L] {x y : L} (hy : IsAlgebraic K y) (h_ev : (Polynomial.aeval x) (minpoly K y) = 0) : ∃ (σ : Gal(L/K)), σ x = y

If x : L is a root of minpoly K y, then we can find (σ : Gal(L/K)) with σ x = y. That is, x and y are Galois conjugates.

theorem minpoly.exists_algEquiv_of_root' {K : Type u_6} {L : Type u_7} [Field K] [Field L] [Algebra K L] [Normal K L] {x y : L} (hy : IsAlgebraic K y) (h_ev : (Polynomial.aeval x) (minpoly K y) = 0) : ∃ (σ : Gal(L/K)), σ y = x

If x : L is a root of minpoly K y, then we can find (σ : Gal(L/K)) with σ y = x. That is, x and y are Galois conjugates.

instance Algebra.IsQuadraticExtension.normal (F : Type u_6) (K : Type u_7) [Field F] [Field K] [Algebra F K] [IsQuadraticExtension F K] : Normal F K

A quadratic extension is normal.