Documentation

Mathlib.Algebra.Polynomial.FieldDivision

Theory of univariate polynomials #

This file starts looking like the ring theory of $R[X]$

theorem Polynomial.rootMultiplicity_sub_one_le_derivative_rootMultiplicity_of_ne_zero {R : Type u} [CommRing R] (p : Polynomial R) (t : R) (hnezero : Polynomial.derivative p 0) :
theorem Polynomial.isRoot_iterate_derivative_of_lt_rootMultiplicity {R : Type u} [CommRing R] {p : Polynomial R} {t : R} {n : } (hn : n < Polynomial.rootMultiplicity t p) :
((Polynomial.derivative)^[n] p).IsRoot t
theorem Polynomial.eval_iterate_derivative_rootMultiplicity {R : Type u} [CommRing R] {p : Polynomial R} {t : R} :
Polynomial.eval t ((Polynomial.derivative)^[Polynomial.rootMultiplicity t p] p) = (Polynomial.rootMultiplicity t p).factorial Polynomial.eval t (p /ₘ (Polynomial.X - Polynomial.C t) ^ Polynomial.rootMultiplicity t p)
theorem Polynomial.lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors {R : Type u} [CommRing R] {p : Polynomial R} {t : R} {n : } (h : p 0) (hroot : mn, ((Polynomial.derivative)^[m] p).IsRoot t) (hnzd : n.factorial nonZeroDivisors R) :
theorem Polynomial.lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors' {R : Type u} [CommRing R] {p : Polynomial R} {t : R} {n : } (h : p 0) (hroot : mn, ((Polynomial.derivative)^[m] p).IsRoot t) (hnzd : mn, m 0m nonZeroDivisors R) :
theorem Polynomial.lt_rootMultiplicity_iff_isRoot_iterate_derivative_of_mem_nonZeroDivisors {R : Type u} [CommRing R] {p : Polynomial R} {t : R} {n : } (h : p 0) (hnzd : n.factorial nonZeroDivisors R) :
n < Polynomial.rootMultiplicity t p mn, ((Polynomial.derivative)^[m] p).IsRoot t
theorem Polynomial.lt_rootMultiplicity_iff_isRoot_iterate_derivative_of_mem_nonZeroDivisors' {R : Type u} [CommRing R] {p : Polynomial R} {t : R} {n : } (h : p 0) (hnzd : mn, m 0m nonZeroDivisors R) :
n < Polynomial.rootMultiplicity t p mn, ((Polynomial.derivative)^[m] p).IsRoot t
theorem Polynomial.one_lt_rootMultiplicity_iff_isRoot_iterate_derivative {R : Type u} [CommRing R] {p : Polynomial R} {t : R} (h : p 0) :
1 < Polynomial.rootMultiplicity t p m1, ((Polynomial.derivative)^[m] p).IsRoot t
theorem Polynomial.one_lt_rootMultiplicity_iff_isRoot {R : Type u} [CommRing R] {p : Polynomial R} {t : R} (h : p 0) :
1 < Polynomial.rootMultiplicity t p p.IsRoot t (Polynomial.derivative p).IsRoot t
theorem Polynomial.one_lt_rootMultiplicity_iff_isRoot_gcd {R : Type u} [CommRing R] [IsDomain R] [GCDMonoid (Polynomial R)] {p : Polynomial R} {t : R} (h : p 0) :
1 < Polynomial.rootMultiplicity t p (gcd p (Polynomial.derivative p)).IsRoot t
theorem Polynomial.derivative_rootMultiplicity_of_root {R : Type u} [CommRing R] [IsDomain R] [CharZero R] {p : Polynomial R} {t : R} (hpt : p.IsRoot t) :
theorem Polynomial.lt_rootMultiplicity_of_isRoot_iterate_derivative {R : Type u} [CommRing R] [IsDomain R] [CharZero R] {p : Polynomial R} {t : R} {n : } (h : p 0) (hroot : mn, ((Polynomial.derivative)^[m] p).IsRoot t) :
theorem Polynomial.lt_rootMultiplicity_iff_isRoot_iterate_derivative {R : Type u} [CommRing R] [IsDomain R] [CharZero R] {p : Polynomial R} {t : R} {n : } (h : p 0) :
n < Polynomial.rootMultiplicity t p mn, ((Polynomial.derivative)^[m] p).IsRoot t
Equations
  • One or more equations did not get rendered due to their size.
@[simp]
theorem Polynomial.coe_normUnit {R : Type u} [CommRing R] [IsDomain R] [NormalizationMonoid R] {p : Polynomial R} :
(normUnit p) = Polynomial.C (normUnit p.leadingCoeff)
theorem Polynomial.leadingCoeff_normalize {R : Type u} [CommRing R] [IsDomain R] [NormalizationMonoid R] (p : Polynomial R) :
(normalize p).leadingCoeff = normalize p.leadingCoeff
theorem Polynomial.Monic.normalize_eq_self {R : Type u} [CommRing R] [IsDomain R] [NormalizationMonoid R] {p : Polynomial R} (hp : p.Monic) :
normalize p = p
theorem Polynomial.roots_normalize {R : Type u} [CommRing R] [IsDomain R] [NormalizationMonoid R] {p : Polynomial R} :
(normalize p).roots = p.roots
theorem Polynomial.normUnit_X {R : Type u} [CommRing R] [IsDomain R] [NormalizationMonoid R] :
normUnit Polynomial.X = 1
theorem Polynomial.X_eq_normalize {R : Type u} [CommRing R] [IsDomain R] [NormalizationMonoid R] :
Polynomial.X = normalize Polynomial.X
theorem Polynomial.degree_pos_of_ne_zero_of_nonunit {R : Type u} [DivisionRing R] {p : Polynomial R} (hp0 : p 0) (hp : ¬IsUnit p) :
0 < p.degree
@[simp]
theorem Polynomial.map_eq_zero {R : Type u} {S : Type v} [DivisionRing R] {p : Polynomial R} [Semiring S] [Nontrivial S] (f : R →+* S) :
Polynomial.map f p = 0 p = 0
theorem Polynomial.map_ne_zero {R : Type u} {S : Type v} [DivisionRing R] {p : Polynomial R} [Semiring S] [Nontrivial S] {f : R →+* S} (hp : p 0) :
@[simp]
theorem Polynomial.degree_map {R : Type u} {S : Type v} [DivisionRing R] [Semiring S] [Nontrivial S] (p : Polynomial R) (f : R →+* S) :
(Polynomial.map f p).degree = p.degree
@[simp]
theorem Polynomial.natDegree_map {R : Type u} {S : Type v} [DivisionRing R] {p : Polynomial R} [Semiring S] [Nontrivial S] (f : R →+* S) :
(Polynomial.map f p).natDegree = p.natDegree
@[simp]
theorem Polynomial.leadingCoeff_map {R : Type u} {S : Type v} [DivisionRing R] {p : Polynomial R} [Semiring S] [Nontrivial S] (f : R →+* S) :
(Polynomial.map f p).leadingCoeff = f p.leadingCoeff
theorem Polynomial.monic_map_iff {R : Type u} {S : Type v} [DivisionRing R] [Semiring S] [Nontrivial S] {f : R →+* S} {p : Polynomial R} :
(Polynomial.map f p).Monic p.Monic
theorem Polynomial.isUnit_iff_degree_eq_zero {R : Type u} [Field R] {p : Polynomial R} :
IsUnit p p.degree = 0
def Polynomial.div {R : Type u} [Field R] (p : Polynomial R) (q : Polynomial R) :

Division of polynomials. See Polynomial.divByMonic for more details.

Equations
  • p.div q = Polynomial.C q.leadingCoeff⁻¹ * (p /ₘ (q * Polynomial.C q.leadingCoeff⁻¹))
Instances For
    def Polynomial.mod {R : Type u} [Field R] (p : Polynomial R) (q : Polynomial R) :

    Remainder of polynomial division. See Polynomial.modByMonic for more details.

    Equations
    • p.mod q = p %ₘ (q * Polynomial.C q.leadingCoeff⁻¹)
    Instances For
      instance Polynomial.instDiv {R : Type u} [Field R] :
      Equations
      • Polynomial.instDiv = { div := Polynomial.div }
      instance Polynomial.instMod {R : Type u} [Field R] :
      Equations
      • Polynomial.instMod = { mod := Polynomial.mod }
      theorem Polynomial.div_def {R : Type u} [Field R] {p : Polynomial R} {q : Polynomial R} :
      p / q = Polynomial.C q.leadingCoeff⁻¹ * (p /ₘ (q * Polynomial.C q.leadingCoeff⁻¹))
      theorem Polynomial.mod_def {R : Type u} [Field R] {p : Polynomial R} {q : Polynomial R} :
      p % q = p %ₘ (q * Polynomial.C q.leadingCoeff⁻¹)
      theorem Polynomial.modByMonic_eq_mod {R : Type u} [Field R] {q : Polynomial R} (p : Polynomial R) (hq : q.Monic) :
      p %ₘ q = p % q
      theorem Polynomial.divByMonic_eq_div {R : Type u} [Field R] {q : Polynomial R} (p : Polynomial R) (hq : q.Monic) :
      p /ₘ q = p / q
      theorem Polynomial.mod_X_sub_C_eq_C_eval {R : Type u} [Field R] (p : Polynomial R) (a : R) :
      p % (Polynomial.X - Polynomial.C a) = Polynomial.C (Polynomial.eval a p)
      theorem Polynomial.mul_div_eq_iff_isRoot {R : Type u} {a : R} [Field R] {p : Polynomial R} :
      (Polynomial.X - Polynomial.C a) * (p / (Polynomial.X - Polynomial.C a)) = p p.IsRoot a
      Equations
      • One or more equations did not get rendered due to their size.
      theorem Polynomial.mod_eq_self_iff {R : Type u} [Field R] {p : Polynomial R} {q : Polynomial R} (hq0 : q 0) :
      p % q = p p.degree < q.degree
      theorem Polynomial.div_eq_zero_iff {R : Type u} [Field R] {p : Polynomial R} {q : Polynomial R} (hq0 : q 0) :
      p / q = 0 p.degree < q.degree
      theorem Polynomial.degree_add_div {R : Type u} [Field R] {p : Polynomial R} {q : Polynomial R} (hq0 : q 0) (hpq : q.degree p.degree) :
      q.degree + (p / q).degree = p.degree
      theorem Polynomial.degree_div_le {R : Type u} [Field R] (p : Polynomial R) (q : Polynomial R) :
      (p / q).degree p.degree
      theorem Polynomial.degree_div_lt {R : Type u} [Field R] {p : Polynomial R} {q : Polynomial R} (hp : p 0) (hq : 0 < q.degree) :
      (p / q).degree < p.degree
      theorem Polynomial.isUnit_map {R : Type u} {k : Type y} [Field R] {p : Polynomial R} [Field k] (f : R →+* k) :
      theorem Polynomial.map_div {R : Type u} {k : Type y} [Field R] {p : Polynomial R} {q : Polynomial R} [Field k] (f : R →+* k) :
      theorem Polynomial.map_mod {R : Type u} {k : Type y} [Field R] {p : Polynomial R} {q : Polynomial R} [Field k] (f : R →+* k) :
      theorem Polynomial.eval₂_gcd_eq_zero {R : Type u} {k : Type y} [Field R] [CommSemiring k] [DecidableEq R] {ϕ : R →+* k} {f : Polynomial R} {g : Polynomial R} {α : k} (hf : Polynomial.eval₂ ϕ α f = 0) (hg : Polynomial.eval₂ ϕ α g = 0) :
      theorem Polynomial.eval_gcd_eq_zero {R : Type u} [Field R] [DecidableEq R] {f : Polynomial R} {g : Polynomial R} {α : R} (hf : Polynomial.eval α f = 0) (hg : Polynomial.eval α g = 0) :
      theorem Polynomial.root_left_of_root_gcd {R : Type u} {k : Type y} [Field R] [CommSemiring k] [DecidableEq R] {ϕ : R →+* k} {f : Polynomial R} {g : Polynomial R} {α : k} (hα : Polynomial.eval₂ ϕ α (EuclideanDomain.gcd f g) = 0) :
      theorem Polynomial.root_right_of_root_gcd {R : Type u} {k : Type y} [Field R] [CommSemiring k] [DecidableEq R] {ϕ : R →+* k} {f : Polynomial R} {g : Polynomial R} {α : k} (hα : Polynomial.eval₂ ϕ α (EuclideanDomain.gcd f g) = 0) :
      theorem Polynomial.isRoot_gcd_iff_isRoot_left_right {R : Type u} [Field R] [DecidableEq R] {f : Polynomial R} {g : Polynomial R} {α : R} :
      (EuclideanDomain.gcd f g).IsRoot α f.IsRoot α g.IsRoot α
      theorem Polynomial.isCoprime_map {R : Type u} {k : Type y} [Field R] {p : Polynomial R} {q : Polynomial R} [Field k] (f : R →+* k) :
      theorem Polynomial.mem_roots_map {R : Type u} {k : Type y} [Field R] {p : Polynomial R} [CommRing k] [IsDomain k] {f : R →+* k} {x : k} (hp : p 0) :
      x (Polynomial.map f p).roots Polynomial.eval₂ f x p = 0
      theorem Polynomial.rootSet_monomial {R : Type u} {S : Type v} [Field R] [CommRing S] [IsDomain S] [Algebra R S] {n : } (hn : n 0) {a : R} (ha : a 0) :
      ((Polynomial.monomial n) a).rootSet S = {0}
      theorem Polynomial.rootSet_C_mul_X_pow {R : Type u} {S : Type v} [Field R] [CommRing S] [IsDomain S] [Algebra R S] {n : } (hn : n 0) {a : R} (ha : a 0) :
      (Polynomial.C a * Polynomial.X ^ n).rootSet S = {0}
      theorem Polynomial.rootSet_X_pow {R : Type u} {S : Type v} [Field R] [CommRing S] [IsDomain S] [Algebra R S] {n : } (hn : n 0) :
      (Polynomial.X ^ n).rootSet S = {0}
      theorem Polynomial.rootSet_prod {R : Type u} {S : Type v} [Field R] [CommRing S] [IsDomain S] [Algebra R S] {ι : Type u_1} (f : ιPolynomial R) (s : Finset ι) (h : s.prod f 0) :
      (s.prod f).rootSet S = is, (f i).rootSet S
      theorem Polynomial.exists_root_of_degree_eq_one {R : Type u} [Field R] {p : Polynomial R} (h : p.degree = 1) :
      ∃ (x : R), p.IsRoot x
      theorem Polynomial.coeff_inv_units {R : Type u} [Field R] (u : (Polynomial R)ˣ) (n : ) :
      ((u).coeff n)⁻¹ = (u⁻¹).coeff n
      theorem Polynomial.monic_normalize {R : Type u} [Field R] {p : Polynomial R} [DecidableEq R] (hp0 : p 0) :
      (normalize p).Monic
      theorem Polynomial.leadingCoeff_div {R : Type u} [Field R] {p : Polynomial R} {q : Polynomial R} (hpq : q.degree p.degree) :
      (p / q).leadingCoeff = p.leadingCoeff / q.leadingCoeff
      theorem Polynomial.div_C_mul {R : Type u} {a : R} [Field R] {p : Polynomial R} {q : Polynomial R} :
      p / (Polynomial.C a * q) = Polynomial.C a⁻¹ * (p / q)
      theorem Polynomial.C_mul_dvd {R : Type u} {a : R} [Field R] {p : Polynomial R} {q : Polynomial R} (ha : a 0) :
      Polynomial.C a * p q p q
      theorem Polynomial.dvd_C_mul {R : Type u} {a : R} [Field R] {p : Polynomial R} {q : Polynomial R} (ha : a 0) :
      p Polynomial.C a * q p q
      theorem Polynomial.coe_normUnit_of_ne_zero {R : Type u} [Field R] {p : Polynomial R} [DecidableEq R] (hp : p 0) :
      (normUnit p) = Polynomial.C p.leadingCoeff⁻¹
      theorem Polynomial.normalize_monic {R : Type u} [Field R] {p : Polynomial R} [DecidableEq R] (h : p.Monic) :
      normalize p = p
      theorem Polynomial.map_dvd_map' {R : Type u} {k : Type y} [Field R] [Field k] (f : R →+* k) {x : Polynomial R} {y : Polynomial R} :
      theorem Polynomial.degree_normalize {R : Type u} [Field R] {p : Polynomial R} [DecidableEq R] :
      (normalize p).degree = p.degree
      theorem Polynomial.prime_of_degree_eq_one {R : Type u} [Field R] {p : Polynomial R} (hp1 : p.degree = 1) :
      theorem Polynomial.irreducible_of_degree_eq_one {R : Type u} [Field R] {p : Polynomial R} (hp1 : p.degree = 1) :
      theorem Polynomial.not_irreducible_C {R : Type u} [Field R] (x : R) :
      ¬Irreducible (Polynomial.C x)
      theorem Polynomial.degree_pos_of_irreducible {R : Type u} [Field R] {p : Polynomial R} (hp : Irreducible p) :
      0 < p.degree
      theorem Polynomial.X_sub_C_mul_divByMonic_eq_sub_modByMonic {K : Type u_1} [Field K] (f : Polynomial K) (a : K) :
      (Polynomial.X - Polynomial.C a) * (f /ₘ (Polynomial.X - Polynomial.C a)) = f - f %ₘ (Polynomial.X - Polynomial.C a)
      theorem Polynomial.divByMonic_add_X_sub_C_mul_derivate_divByMonic_eq_derivative {K : Type u_1} [Field K] (f : Polynomial K) (a : K) :
      f /ₘ (Polynomial.X - Polynomial.C a) + (Polynomial.X - Polynomial.C a) * Polynomial.derivative (f /ₘ (Polynomial.X - Polynomial.C a)) = Polynomial.derivative f
      theorem Polynomial.X_sub_C_dvd_derivative_of_X_sub_C_dvd_divByMonic {K : Type u_1} [Field K] (f : Polynomial K) {a : K} (hf : Polynomial.X - Polynomial.C a f /ₘ (Polynomial.X - Polynomial.C a)) :
      Polynomial.X - Polynomial.C a Polynomial.derivative f
      theorem Polynomial.isCoprime_of_is_root_of_eval_derivative_ne_zero {K : Type u_1} [Field K] (f : Polynomial K) (a : K) (hf' : Polynomial.eval a (Polynomial.derivative f) 0) :
      IsCoprime (Polynomial.X - Polynomial.C a) (f /ₘ (Polynomial.X - Polynomial.C a))

      If f is a polynomial over a field, and a : K satisfies f' a ≠ 0, then f / (X - a) is coprime with X - a. Note that we do not assume f a = 0, because f / (X - a) = (f - f a) / (X - a).

      theorem Polynomial.irreducible_iff_degree_lt {R : Type u} [Field R] (p : Polynomial R) (hp0 : p 0) (hpu : ¬IsUnit p) :
      Irreducible p ∀ (q : Polynomial R), q.degree (p.natDegree / 2)q pIsUnit q

      To check a polynomial over a field is irreducible, it suffices to check only for divisors that have smaller degree.

      See also: Polynomial.Monic.irreducible_iff_natDegree.

      theorem Polynomial.irreducible_iff_lt_natDegree_lt {R : Type u} [Field R] {p : Polynomial R} (hp0 : p 0) (hpu : ¬IsUnit p) :
      Irreducible p ∀ (q : Polynomial R), q.Monicq.natDegree Finset.Ioc 0 (p.natDegree / 2)¬q p

      To check a polynomial p over a field is irreducible, it suffices to check there are no divisors of degree 0 < d ≤ degree p / 2.

      See also: Polynomial.Monic.irreducible_iff_natDegree'.

      theorem Irreducible.natDegree_pos {F : Type u_1} [Field F] {f : Polynomial F} (h : Irreducible f) :
      0 < f.natDegree

      An irreducible polynomial over a field must have positive degree.