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Mathlib.FieldTheory.Separable

Separable polynomials #

We define a polynomial to be separable if it is coprime with its derivative. We prove basic properties about separable polynomials here.

Main definitions #

A polynomial is separable iff it is coprime with its derivative.

Equations
Instances For
    theorem Polynomial.separable_def {R : Type u} [CommSemiring R] (f : Polynomial R) :
    Polynomial.Separable f IsCoprime f (Polynomial.derivative f)
    theorem Polynomial.separable_def' {R : Type u} [CommSemiring R] (f : Polynomial R) :
    Polynomial.Separable f a b, a * f + b * Polynomial.derivative f = 1
    theorem Polynomial.separable_X_add_C {R : Type u} [CommSemiring R] (a : R) :
    Polynomial.Separable (Polynomial.X + Polynomial.C a)
    theorem Polynomial.separable_C {R : Type u} [CommSemiring R] (r : R) :
    Polynomial.Separable (Polynomial.C r) IsUnit r
    theorem Polynomial.Separable.of_pow {R : Type u} [CommSemiring R] {f : Polynomial R} (hf : ¬IsUnit f) {n : } (hn : n 0) (hfs : Polynomial.Separable (f ^ n)) :
    theorem Polynomial.separable_X_sub_C {R : Type u} [CommRing R] {x : R} :
    Polynomial.Separable (Polynomial.X - Polynomial.C x)
    theorem Polynomial.separable_prod' {R : Type u} [CommRing R] {ι : Type u_1} {f : ιPolynomial R} {s : Finset ι} :
    (∀ (x : ι), x s∀ (y : ι), y sx yIsCoprime (f x) (f y)) → (∀ (x : ι), x sPolynomial.Separable (f x)) → Polynomial.Separable (Finset.prod s fun x => f x)
    theorem Polynomial.separable_prod {R : Type u} [CommRing R] {ι : Type u_1} [Fintype ι] {f : ιPolynomial R} (h1 : Pairwise (IsCoprime on f)) (h2 : ∀ (x : ι), Polynomial.Separable (f x)) :
    Polynomial.Separable (Finset.prod Finset.univ fun x => f x)
    theorem Polynomial.Separable.inj_of_prod_X_sub_C {R : Type u} [CommRing R] [Nontrivial R] {ι : Type u_1} {f : ιR} {s : Finset ι} (hfs : Polynomial.Separable (Finset.prod s fun i => Polynomial.X - Polynomial.C (f i))) {x : ι} {y : ι} (hx : x s) (hy : y s) (hfxy : f x = f y) :
    x = y
    theorem Polynomial.Separable.injective_of_prod_X_sub_C {R : Type u} [CommRing R] [Nontrivial R] {ι : Type u_1} [Fintype ι] {f : ιR} (hfs : Polynomial.Separable (Finset.prod Finset.univ fun i => Polynomial.X - Polynomial.C (f i))) :
    theorem Polynomial.nodup_of_separable_prod {R : Type u} [CommRing R] [Nontrivial R] {s : Multiset R} (hs : Polynomial.Separable (Multiset.prod (Multiset.map (fun a => Polynomial.X - Polynomial.C a) s))) :
    theorem Polynomial.separable_X_pow_sub_C_unit {R : Type u} [CommRing R] {n : } (u : Rˣ) (hn : IsUnit n) :
    Polynomial.Separable (Polynomial.X ^ n - Polynomial.C u)

    If IsUnit n in a CommRing R, then X ^ n - u is separable for any unit u.

    theorem Polynomial.separable_iff_derivative_ne_zero {F : Type u} [Field F] {f : Polynomial F} (hf : Irreducible f) :
    Polynomial.Separable f Polynomial.derivative f 0
    theorem Polynomial.separable_prod_X_sub_C_iff' {F : Type u} [Field F] {ι : Type u_1} {f : ιF} {s : Finset ι} :
    Polynomial.Separable (Finset.prod s fun i => Polynomial.X - Polynomial.C (f i)) ∀ (x : ι), x s∀ (y : ι), y sf x = f yx = y
    theorem Polynomial.separable_prod_X_sub_C_iff {F : Type u} [Field F] {ι : Type u_1} [Fintype ι] {f : ιF} :
    Polynomial.Separable (Finset.prod Finset.univ fun i => Polynomial.X - Polynomial.C (f i)) Function.Injective f
    theorem Polynomial.exists_separable_of_irreducible {F : Type u} [Field F] (p : ) [HF : CharP F p] {f : Polynomial F} (hf : Irreducible f) (hp : p 0) :
    n g, Polynomial.Separable g ↑(Polynomial.expand F (p ^ n)) g = f
    theorem Polynomial.isUnit_or_eq_zero_of_separable_expand {F : Type u} [Field F] (p : ) [HF : CharP F p] {f : Polynomial F} (n : ) (hp : 0 < p) (hf : Polynomial.Separable (↑(Polynomial.expand F (p ^ n)) f)) :
    IsUnit f n = 0
    theorem Polynomial.unique_separable_of_irreducible {F : Type u} [Field F] (p : ) [HF : CharP F p] {f : Polynomial F} (hf : Irreducible f) (hp : 0 < p) (n₁ : ) (g₁ : Polynomial F) (hg₁ : Polynomial.Separable g₁) (hgf₁ : ↑(Polynomial.expand F (p ^ n₁)) g₁ = f) (n₂ : ) (g₂ : Polynomial F) (hg₂ : Polynomial.Separable g₂) (hgf₂ : ↑(Polynomial.expand F (p ^ n₂)) g₂ = f) :
    n₁ = n₂ g₁ = g₂
    theorem Polynomial.separable_X_pow_sub_C {F : Type u} [Field F] {n : } (a : F) (hn : n 0) (ha : a 0) :
    Polynomial.Separable (Polynomial.X ^ n - Polynomial.C a)

    If n ≠ 0 in F, then X ^ n - a is separable for any a ≠ 0.

    theorem Polynomial.X_pow_sub_one_separable_iff {F : Type u} [Field F] {n : } :
    Polynomial.Separable (Polynomial.X ^ n - 1) n 0

    In a field F, X ^ n - 1 is separable iff ↑n ≠ 0.

    theorem Polynomial.eq_X_sub_C_of_separable_of_root_eq {F : Type u} [Field F] {K : Type v} [Field K] {i : F →+* K} {x : F} {h : Polynomial F} (h_sep : Polynomial.Separable h) (h_root : Polynomial.eval x h = 0) (h_splits : Polynomial.Splits i h) (h_roots : ∀ (y : K), y Polynomial.roots (Polynomial.map i h)y = i x) :
    h = Polynomial.C (Polynomial.leadingCoeff h) * (Polynomial.X - Polynomial.C x)
    theorem Polynomial.exists_finset_of_splits {F : Type u} [Field F] {K : Type v} [Field K] (i : F →+* K) {f : Polynomial F} (sep : Polynomial.Separable f) (sp : Polynomial.Splits i f) :
    s, Polynomial.map i f = Polynomial.C (i (Polynomial.leadingCoeff f)) * Finset.prod s fun a => Polynomial.X - Polynomial.C a
    class IsSeparable (F : Type u_1) (K : Type u_2) [CommRing F] [Ring K] [Algebra F K] :

    Typeclass for separable field extension: K is a separable field extension of F iff the minimal polynomial of every x : K is separable.

    We define this for general (commutative) rings and only assume F and K are fields if this is needed for a proof.

    Instances
      theorem IsSeparable.isIntegral (F : Type u_3) {K : Type u_4} [CommRing F] [Ring K] [Algebra F K] [IsSeparable F K] (x : K) :
      theorem IsSeparable.separable (F : Type u_3) {K : Type u_4} [CommRing F] [Ring K] [Algebra F K] [IsSeparable F K] (x : K) :
      theorem isSeparable_iff {F : Type u_5} {K : Type u_6} [CommRing F] [Ring K] [Algebra F K] :
      instance IsSeparable.of_finite (F : Type u_1) (K : Type u_2) [Field F] [Field K] [Algebra F K] [FiniteDimensional F K] [CharZero F] :

      A finite field extension in characteristic 0 is separable.

      Equations
      theorem isSeparable_tower_top_of_isSeparable (F : Type u_1) (K : Type u_2) (E : Type u_3) [Field F] [Field K] [Field E] [Algebra F K] [Algebra F E] [Algebra K E] [IsScalarTower F K E] [IsSeparable F E] :
      theorem isSeparable_tower_bot_of_isSeparable (F : Type u_1) (K : Type u_2) (E : Type u_3) [Field F] [Field K] [Field E] [Algebra F K] [Algebra F E] [Algebra K E] [IsScalarTower F K E] [h : IsSeparable F E] :
      theorem IsSeparable.of_algHom (F : Type u_1) {E : Type u_3} [Field F] [Field E] [Algebra F E] (E' : Type u_4) [Field E'] [Algebra F E'] (f : E →ₐ[F] E') [IsSeparable F E'] :
      theorem AlgHom.card_of_powerBasis {S : Type u_2} [CommRing S] {K : Type u_4} {L : Type u_5} [Field K] [Field L] [Algebra K S] [Algebra K L] (pb : PowerBasis K S) (h_sep : Polynomial.Separable (minpoly K pb.gen)) (h_splits : Polynomial.Splits (algebraMap K L) (minpoly K pb.gen)) :
      Fintype.card (S →ₐ[K] L) = pb.dim