Documentation

Mathlib.Data.Polynomial.Splits

Split polynomials #

A polynomial f : K[X] splits over a field extension L of K if it is zero or all of its irreducible factors over L have degree 1.

Main definitions #

def Polynomial.Splits {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) (f : Polynomial K) :

A polynomial Splits iff it is zero or all of its irreducible factors have degree 1.

Equations
Instances For
    @[simp]
    theorem Polynomial.splits_zero {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) :
    theorem Polynomial.splits_of_map_eq_C {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f : Polynomial K} {a : L} (h : Polynomial.map i f = Polynomial.C a) :
    @[simp]
    theorem Polynomial.splits_C {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) (a : K) :
    Polynomial.Splits i (Polynomial.C a)
    theorem Polynomial.splits_of_degree_eq_one {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f : Polynomial K} (hf : Polynomial.degree f = 1) :
    theorem Polynomial.splits_mul {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f : Polynomial K} {g : Polynomial K} (hf : Polynomial.Splits i f) (hg : Polynomial.Splits i g) :
    theorem Polynomial.splits_of_splits_mul' {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f : Polynomial K} {g : Polynomial K} (hfg : Polynomial.map i (f * g) 0) (h : Polynomial.Splits i (f * g)) :
    theorem Polynomial.splits_map_iff {F : Type u} {K : Type v} {L : Type w} [CommRing K] [Field L] [Field F] (i : K →+* L) (j : L →+* F) {f : Polynomial K} :
    theorem Polynomial.splits_one {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) :
    theorem Polynomial.splits_of_isUnit {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) [IsDomain K] {u : Polynomial K} (hu : IsUnit u) :
    theorem Polynomial.splits_X_sub_C {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {x : K} :
    Polynomial.Splits i (Polynomial.X - Polynomial.C x)
    theorem Polynomial.splits_X {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) :
    Polynomial.Splits i Polynomial.X
    theorem Polynomial.splits_prod {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {ι : Type u} {s : ιPolynomial K} {t : Finset ι} :
    (jt, Polynomial.Splits i (s j))Polynomial.Splits i (Finset.prod t fun (x : ι) => s x)
    theorem Polynomial.splits_pow {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f : Polynomial K} (hf : Polynomial.Splits i f) (n : ) :
    theorem Polynomial.splits_X_pow {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) (n : ) :
    Polynomial.Splits i (Polynomial.X ^ n)
    theorem Polynomial.exists_root_of_splits' {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f : Polynomial K} (hs : Polynomial.Splits i f) (hf0 : Polynomial.degree (Polynomial.map i f) 0) :
    ∃ (x : L), Polynomial.eval₂ i x f = 0
    def Polynomial.rootOfSplits' {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f : Polynomial K} (hf : Polynomial.Splits i f) (hfd : Polynomial.degree (Polynomial.map i f) 0) :
    L

    Pick a root of a polynomial that splits. See rootOfSplits for polynomials over a field which has simpler assumptions.

    Equations
    Instances For
      theorem Polynomial.natDegree_eq_card_roots' {K : Type v} {L : Type w} [CommRing K] [Field L] {p : Polynomial K} {i : K →+* L} (hsplit : Polynomial.Splits i p) :
      theorem Polynomial.degree_eq_card_roots' {K : Type v} {L : Type w} [CommRing K] [Field L] {p : Polynomial K} {i : K →+* L} (p_ne_zero : Polynomial.map i p 0) (hsplit : Polynomial.Splits i p) :
      theorem Polynomial.splits_iff {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) (f : Polynomial K) :

      This lemma is for polynomials over a field.

      theorem Polynomial.Splits.def' {K : Type v} {L : Type w} [Field K] [Field L] {i : K →+* L} {f : Polynomial K} (h : Polynomial.Splits i f) :
      f = 0 ∀ {g : Polynomial L}, Irreducible gg Polynomial.map i fPolynomial.degree g = 1

      This lemma is for polynomials over a field.

      theorem Polynomial.splits_of_splits_mul {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} {g : Polynomial K} (hfg : f * g 0) (h : Polynomial.Splits i (f * g)) :
      theorem Polynomial.splits_of_splits_of_dvd {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} {g : Polynomial K} (hf0 : f 0) (hf : Polynomial.Splits i f) (hgf : g f) :
      theorem Polynomial.splits_of_splits_gcd_left {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) [DecidableEq K] {f : Polynomial K} {g : Polynomial K} (hf0 : f 0) (hf : Polynomial.Splits i f) :
      theorem Polynomial.splits_of_splits_gcd_right {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) [DecidableEq K] {f : Polynomial K} {g : Polynomial K} (hg0 : g 0) (hg : Polynomial.Splits i g) :
      theorem Polynomial.splits_mul_iff {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} {g : Polynomial K} (hf : f 0) (hg : g 0) :
      theorem Polynomial.splits_prod_iff {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {ι : Type u} {s : ιPolynomial K} {t : Finset ι} :
      (jt, s j 0)(Polynomial.Splits i (Finset.prod t fun (x : ι) => s x) jt, Polynomial.Splits i (s j))
      theorem Polynomial.exists_root_of_splits {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} (hs : Polynomial.Splits i f) (hf0 : Polynomial.degree f 0) :
      ∃ (x : L), Polynomial.eval₂ i x f = 0
      def Polynomial.rootOfSplits {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} (hf : Polynomial.Splits i f) (hfd : Polynomial.degree f 0) :
      L

      Pick a root of a polynomial that splits. This version is for polynomials over a field and has simpler assumptions.

      Equations
      Instances For

        rootOfSplits' is definitionally equal to rootOfSplits.

        theorem Polynomial.map_rootOfSplits {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} (hf : Polynomial.Splits i f) (hfd : Polynomial.degree f 0) :
        theorem Polynomial.natDegree_eq_card_roots {K : Type v} {L : Type w} [Field K] [Field L] {p : Polynomial K} {i : K →+* L} (hsplit : Polynomial.Splits i p) :
        theorem Polynomial.degree_eq_card_roots {K : Type v} {L : Type w} [Field K] [Field L] {p : Polynomial K} {i : K →+* L} (p_ne_zero : p 0) (hsplit : Polynomial.Splits i p) :
        Polynomial.degree p = (Multiset.card (Polynomial.roots (Polynomial.map i p)))
        theorem Polynomial.image_rootSet {R : Type u_1} {K : Type v} {L : Type w} [CommRing R] [Field K] [Field L] [Algebra R K] [Algebra R L] {p : Polynomial R} (h : Polynomial.Splits (algebraMap R K) p) (f : K →ₐ[R] L) :
        theorem Polynomial.eq_prod_roots_of_splits {K : Type v} {L : Type w} [Field K] [Field L] {p : Polynomial K} {i : K →+* L} (hsplit : Polynomial.Splits i p) :
        Polynomial.map i p = Polynomial.C (i (Polynomial.leadingCoeff p)) * Multiset.prod (Multiset.map (fun (a : L) => Polynomial.X - Polynomial.C a) (Polynomial.roots (Polynomial.map i p)))
        theorem Polynomial.eq_prod_roots_of_splits_id {K : Type v} [Field K] {p : Polynomial K} (hsplit : Polynomial.Splits (RingHom.id K) p) :
        p = Polynomial.C (Polynomial.leadingCoeff p) * Multiset.prod (Multiset.map (fun (a : K) => Polynomial.X - Polynomial.C a) (Polynomial.roots p))
        theorem Polynomial.eq_prod_roots_of_monic_of_splits_id {K : Type v} [Field K] {p : Polynomial K} (m : Polynomial.Monic p) (hsplit : Polynomial.Splits (RingHom.id K) p) :
        p = Multiset.prod (Multiset.map (fun (a : K) => Polynomial.X - Polynomial.C a) (Polynomial.roots p))
        theorem Polynomial.eq_X_sub_C_of_splits_of_single_root {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {x : K} {h : Polynomial K} (h_splits : Polynomial.Splits i h) (h_roots : Polynomial.roots (Polynomial.map i h) = {i x}) :
        h = Polynomial.C (Polynomial.leadingCoeff h) * (Polynomial.X - Polynomial.C x)
        theorem Polynomial.splits_of_exists_multiset {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} {s : Multiset L} (hs : Polynomial.map i f = Polynomial.C (i (Polynomial.leadingCoeff f)) * Multiset.prod (Multiset.map (fun (a : L) => Polynomial.X - Polynomial.C a) s)) :
        theorem Polynomial.splits_iff_exists_multiset {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} :
        Polynomial.Splits i f ∃ (s : Multiset L), Polynomial.map i f = Polynomial.C (i (Polynomial.leadingCoeff f)) * Multiset.prod (Multiset.map (fun (a : L) => Polynomial.X - Polynomial.C a) s)
        theorem Polynomial.splits_of_comp {F : Type u} {K : Type v} {L : Type w} [Field K] [Field L] [Field F] (i : K →+* L) (j : L →+* F) {f : Polynomial K} (h : Polynomial.Splits (RingHom.comp j i) f) (roots_mem_range : aPolynomial.roots (Polynomial.map (RingHom.comp j i) f), a RingHom.range j) :
        theorem Polynomial.splits_id_of_splits {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} (h : Polynomial.Splits i f) (roots_mem_range : aPolynomial.roots (Polynomial.map i f), a RingHom.range i) :
        theorem Polynomial.splits_comp_of_splits {R : Type u_1} {K : Type v} {L : Type w} [CommRing R] [Field K] [Field L] (i : R →+* K) (j : K →+* L) {f : Polynomial R} (h : Polynomial.Splits i f) :
        theorem Polynomial.splits_of_algHom {R : Type u_1} {K : Type v} {L : Type w} [CommRing R] [Field K] [Field L] [Algebra R K] [Algebra R L] {f : Polynomial R} (h : Polynomial.Splits (algebraMap R K) f) (e : K →ₐ[R] L) :
        theorem Polynomial.splits_of_isScalarTower {R : Type u_1} {K : Type v} (L : Type w) [CommRing R] [Field K] [Field L] [Algebra R K] [Algebra R L] {f : Polynomial R} [Algebra K L] [IsScalarTower R K L] (h : Polynomial.Splits (algebraMap R K) f) :

        A polynomial splits if and only if it has as many roots as its degree.

        theorem Polynomial.aeval_root_derivative_of_splits {K : Type v} {L : Type w} [Field K] [Field L] [Algebra K L] [DecidableEq L] {P : Polynomial K} (hmo : Polynomial.Monic P) (hP : Polynomial.Splits (algebraMap K L) P) {r : L} (hr : r Polynomial.aroots P L) :
        (Polynomial.aeval r) (Polynomial.derivative P) = Multiset.prod (Multiset.map (fun (a : L) => r - a) (Multiset.erase (Polynomial.aroots P L) r))

        If P is a monic polynomial that splits, then coeff P 0 equals the product of the roots.

        If P is a monic polynomial that splits, then P.nextCoeff equals the sum of the roots.