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Mathlib.FieldTheory.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_isSplittingField and Normal.exists_isSplittingField).

Main Definitions #

class Normal (F : Type u_1) (K : Type u_2) [Field F] [Field K] [Algebra F K] extends Algebra.IsAlgebraic :

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
    theorem Normal.splits' {F : Type u_1} {K : Type u_2} [Field F] [Field K] [Algebra F K] [self : Normal F K] (x : K) :
    theorem Normal.isIntegral {F : Type u_1} {K : Type u_2} [Field F] [Field K] [Algebra F K] :
    Normal F K∀ (x : K), IsIntegral F x
    theorem Normal.splits {F : Type u_1} {K : Type u_2} [Field F] [Field K] [Algebra F K] :
    Normal F K∀ (x : K), Polynomial.Splits (algebraMap F K) (minpoly F x)
    theorem normal_iff {F : Type u_1} {K : Type u_2} [Field F] [Field K] [Algebra F K] :
    Normal F K ∀ (x : K), IsIntegral F x Polynomial.Splits (algebraMap F K) (minpoly F x)
    theorem Normal.out {F : Type u_1} {K : Type u_2} [Field F] [Field K] [Algebra F K] :
    Normal F K∀ (x : K), IsIntegral F x Polynomial.Splits (algebraMap F K) (minpoly F x)
    instance normal_self (F : Type u_1) [Field F] :
    Normal F F
    Equations
    • =
    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] :
    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] [IsScalarTower F K E] [h : Normal F E] :
    Normal K E
    theorem AlgHom.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] [IsScalarTower F K E] [h : Normal F E] (ϕ : E →ₐ[F] K) :
    theorem Normal.of_algEquiv {F : Type u_1} {E : Type u_3} {E' : Type u_4} [Field F] [Field E] [Algebra F E] [Field E'] [Algebra F E'] [h : Normal F E] (f : E ≃ₐ[F] E') :
    Normal F E'
    theorem AlgEquiv.transfer_normal {F : Type u_1} {E : Type u_3} {E' : Type u_4} [Field F] [Field E] [Algebra F E] [Field E'] [Algebra F E'] (f : E ≃ₐ[F] E') :
    Normal F E Normal F E'
    theorem Normal.of_isSplittingField {F : Type u_1} {E : Type u_3} [Field F] [Field E] [Algebra F E] (p : Polynomial F) [hFEp : Polynomial.IsSplittingField F E p] :
    Normal F E
    instance Polynomial.SplittingField.instNormal {F : Type u_1} [Field F] (p : Polynomial F) :
    Normal F p.SplittingField
    Equations
    • =
    instance IntermediateField.normal_iSup (F : Type u_1) (K : Type u_2) [Field F] [Field K] [Algebra F K] {ι : Type u_3} (t : ιIntermediateField F K) [h : ∀ (i : ι), Normal F (t i)] :
    Normal F (⨆ (i : ι), t i)

    A compositum of normal extensions is normal

    Equations
    • =
    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 : xS, IsIntegral F x Polynomial.Splits (algebraMap F L) (minpoly F x)) {x : K} (hx : x IntermediateField.adjoin F S) :

    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 : IntermediateField F K) (E' : IntermediateField F K) [Normal F E] [Normal F E'] :
    Normal F (E E')
    Equations
    • =
    @[simp]
    theorem IntermediateField.restrictScalars_normal {F : Type u_1} {K : Type u_2} {L : Type u_3} [Field F] [Field K] [Field L] [Algebra F L] [Algebra K L] [Algebra F K] [IsScalarTower F K L] {E : IntermediateField K L} :
    def AlgHom.restrictNormalAux {F : Type u_1} {K₁ : Type u_3} {K₂ : Type u_4} [Field F] [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 E K₁] [Algebra E K₂] [IsScalarTower F E K₁] [IsScalarTower F E K₂] [h : Normal F E] :
    (IsScalarTower.toAlgHom F E K₁).range →ₐ[F] (IsScalarTower.toAlgHom F E K₂).range

    Restrict algebra homomorphism to image of normal subfield

    Equations
    • ϕ.restrictNormalAux E = { toFun := fun (x : (IsScalarTower.toAlgHom F E K₁).range) => ϕ x, , map_one' := , map_mul' := , map_zero' := , map_add' := , commutes' := }
    Instances For
      def AlgHom.restrictNormal {F : Type u_1} {K₁ : Type u_3} {K₂ : Type u_4} [Field F] [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 E K₁] [Algebra E K₂] [IsScalarTower F E K₁] [IsScalarTower F E K₂] [Normal F E] :

      Restrict algebra homomorphism to normal subfield

      Equations
      Instances For
        def AlgHom.restrictNormal' {F : Type u_1} {K₁ : Type u_3} {K₂ : Type u_4} [Field F] [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 E K₁] [Algebra E K₂] [IsScalarTower F E K₁] [IsScalarTower F E K₂] [Normal F E] :

        Restrict algebra homomorphism to normal subfield (AlgEquiv version)

        Equations
        Instances For
          @[simp]
          theorem AlgHom.restrictNormal_commutes {F : Type u_1} {K₁ : Type u_3} {K₂ : Type u_4} [Field F] [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 E K₁] [Algebra E K₂] [IsScalarTower F E K₁] [IsScalarTower F E K₂] [Normal F E] (x : E) :
          (algebraMap E K₂) ((ϕ.restrictNormal E) x) = ϕ ((algebraMap E K₁) x)
          theorem AlgHom.restrictNormal_comp {F : Type u_1} {K₁ : Type u_3} {K₂ : Type u_4} {K₃ : Type u_5} [Field F] [Field K₁] [Field K₂] [Field K₃] [Algebra F K₁] [Algebra F K₂] [Algebra F K₃] (ϕ : K₁ →ₐ[F] K₂) (ψ : K₂ →ₐ[F] K₃) (E : Type u_6) [Field E] [Algebra F E] [Algebra E K₁] [Algebra E K₂] [Algebra E K₃] [IsScalarTower F E K₁] [IsScalarTower F E K₂] [IsScalarTower F E K₃] [Normal F E] :
          (ψ.restrictNormal E).comp (ϕ.restrictNormal E) = (ψ.comp ϕ).restrictNormal E
          theorem AlgHom.fieldRange_of_normal {F : Type u_1} {K : Type u_2} [Field K] [Field F] [Algebra F K] {E : IntermediateField F K} [Normal F E] (f : E →ₐ[F] K) :
          f.fieldRange = E
          def AlgEquiv.restrictNormal {F : Type u_1} {K₁ : Type u_3} {K₂ : Type u_4} [Field F] [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 E K₁] [Algebra E K₂] [IsScalarTower F E K₁] [IsScalarTower F E K₂] [Normal F E] :

          Restrict algebra isomorphism to a normal subfield

          Equations
          • χ.restrictNormal E = (χ).restrictNormal' E
          Instances For
            @[simp]
            theorem AlgEquiv.restrictNormal_commutes {F : Type u_1} {K₁ : Type u_3} {K₂ : Type u_4} [Field F] [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 E K₁] [Algebra E K₂] [IsScalarTower F E K₁] [IsScalarTower F E K₂] [Normal F E] (x : E) :
            (algebraMap E K₂) ((χ.restrictNormal E) x) = χ ((algebraMap E K₁) x)
            theorem AlgEquiv.restrictNormal_trans {F : Type u_1} {K₁ : Type u_3} {K₂ : Type u_4} {K₃ : Type u_5} [Field F] [Field K₁] [Field K₂] [Field K₃] [Algebra F K₁] [Algebra F K₂] [Algebra F K₃] (χ : K₁ ≃ₐ[F] K₂) (ω : K₂ ≃ₐ[F] K₃) (E : Type u_6) [Field E] [Algebra F E] [Algebra E K₁] [Algebra E K₂] [Algebra E K₃] [IsScalarTower F E K₁] [IsScalarTower F E K₂] [IsScalarTower F E K₃] [Normal F E] :
            (χ.trans ω).restrictNormal E = (χ.restrictNormal E).trans (ω.restrictNormal E)
            def AlgEquiv.restrictNormalHom {F : Type u_1} {K₁ : Type u_3} [Field F] [Field K₁] [Algebra F K₁] (E : Type u_6) [Field E] [Algebra F E] [Algebra E K₁] [IsScalarTower F E K₁] [Normal F E] :
            (K₁ ≃ₐ[F] K₁) →* E ≃ₐ[F] E

            Restriction to a normal subfield as a group homomorphism

            Equations
            Instances For
              @[simp]
              theorem Normal.algHomEquivAut_apply (F : Type u_1) (K₁ : Type u_3) [Field F] [Field K₁] [Algebra F K₁] (E : Type u_6) [Field E] [Algebra F E] [Algebra E K₁] [IsScalarTower F E K₁] [Normal F E] (σ : E →ₐ[F] K₁) :
              (Normal.algHomEquivAut F K₁ E) σ = σ.restrictNormal' E
              @[simp]
              theorem Normal.algHomEquivAut_symm_apply (F : Type u_1) (K₁ : Type u_3) [Field F] [Field K₁] [Algebra F K₁] (E : Type u_6) [Field E] [Algebra F E] [Algebra E K₁] [IsScalarTower F E K₁] [Normal F E] (σ : E ≃ₐ[F] E) :
              (Normal.algHomEquivAut F K₁ E).symm σ = (IsScalarTower.toAlgHom F E K₁).comp σ
              def Normal.algHomEquivAut (F : Type u_1) (K₁ : Type u_3) [Field F] [Field K₁] [Algebra F K₁] (E : Type u_6) [Field E] [Algebra F E] [Algebra E K₁] [IsScalarTower F E K₁] [Normal F E] :
              (E →ₐ[F] K₁) E ≃ₐ[F] E

              If K₁/E/F is a tower of fields with E/F normal then AlgHom.restrictNormal' is an equivalence.

              Equations
              • One or more equations did not get rendered due to their size.
              Instances For
                noncomputable def AlgHom.liftNormal {F : Type u_1} {K₁ : Type u_3} {K₂ : Type u_4} [Field F] [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] :

                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.

                Equations
                Instances For
                  @[simp]
                  theorem AlgHom.liftNormal_commutes {F : Type u_1} {K₁ : Type u_3} {K₂ : Type u_4} [Field F] [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} {K₁ : Type u_3} [Field F] [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} {K₁ : Type u_3} {K₂ : Type u_4} [Field F] [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] :

                  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 : E ≃ₐ[F] E.

                  Equations
                  Instances For
                    @[simp]
                    theorem AlgEquiv.liftNormal_commutes {F : Type u_1} {K₁ : Type u_3} {K₂ : Type u_4} [Field F] [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} {K₁ : Type u_3} [Field F] [Field K₁] [Algebra F K₁] (E : Type u_6) [Field E] [Algebra F E] [Algebra K₁ E] [IsScalarTower F K₁ E] (χ : K₁ ≃ₐ[F] K₁) [Normal F K₁] [Normal F E] :
                    (χ.liftNormal E).restrictNormal K₁ = χ
                    theorem AlgEquiv.restrictNormalHom_surjective {F : Type u_1} {K₁ : Type u_3} [Field F] [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] :
                    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 : E} {y : E} :
                    theorem isSolvable_of_isScalarTower (F : Type u_1) (K₁ : Type u_3) [Field F] [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 (K₁ ≃ₐ[F] K₁)] [h2 : IsSolvable (E ≃ₐ[K₁] E)] :