Theory of univariate polynomials

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

Main definitions

• Polynomial.roots p: The multiset containing all the roots of p, including their multiplicities.
• Polynomial.rootSet p E: The set of distinct roots of p in an algebra E.

Main statements

• Polynomial.C_leadingCoeff_mul_prod_multiset_X_sub_C: If a polynomial has as many roots as its degree, it can be written as the product of its leading coefficient with ∏ (X - a) where a ranges through its roots.

theorem Polynomial.natDegree_pos_of_aeval_root {R : Type u} {S : Type v} [CommRing R] [Semiring S] [Algebra R S] {p : Polynomial R} (hp : p ≠ 0) {z : S} (hz : (Polynomial.aeval z) p = 0) (inj : ∀ (x : R), (algebraMap R S) x = 0 → x = 0) : 0 < p.natDegree

theorem Polynomial.degree_pos_of_aeval_root {R : Type u} {S : Type v} [CommRing R] [Semiring S] [Algebra R S] {p : Polynomial R} (hp : p ≠ 0) {z : S} (hz : (Polynomial.aeval z) p = 0) (inj : ∀ (x : R), (algebraMap R S) x = 0 → x = 0) : 0 < p.degree

theorem Polynomial.modByMonic_eq_of_dvd_sub {R : Type u} [CommRing R] {q : Polynomial R} (hq : q.Monic) {p₁ : Polynomial R} {p₂ : Polynomial R} (h : q ∣ p₁ - p₂) : p₁.modByMonic q = p₂.modByMonic q

theorem Polynomial.add_modByMonic {R : Type u} [CommRing R] {q : Polynomial R} (p₁ : Polynomial R) (p₂ : Polynomial R) : (p₁ + p₂).modByMonic q = p₁.modByMonic q + p₂.modByMonic q

theorem Polynomial.smul_modByMonic {R : Type u} [CommRing R] {q : Polynomial R} (c : R) (p : Polynomial R) : (c • p).modByMonic q = c • p.modByMonic q

@[simp]

theorem Polynomial.modByMonicHom_apply {R : Type u} [CommRing R] (q : Polynomial R) (p : Polynomial R) : q.modByMonicHom p = p.modByMonic q

def Polynomial.modByMonicHom {R : Type u} [CommRing R] (q : Polynomial R) : Polynomial R →ₗ[R] Polynomial R

_ %ₘ q as an R-linear map.

Equations

• q.modByMonicHom = { toFun := fun (p : Polynomial R) => p.modByMonic q, map_add' := ⋯, map_smul' := ⋯ }

theorem Polynomial.neg_modByMonic {R : Type u} [CommRing R] (p : Polynomial R) (mod : Polynomial R) : (-p).modByMonic mod = -p.modByMonic mod

theorem Polynomial.sub_modByMonic {R : Type u} [CommRing R] (a : Polynomial R) (b : Polynomial R) (mod : Polynomial R) : (a - b).modByMonic mod = a.modByMonic mod - b.modByMonic mod

theorem Polynomial.aeval_modByMonic_eq_self_of_root {R : Type u} {S : Type v} [CommRing R] [Ring S] [Algebra R S] {p : Polynomial R} {q : Polynomial R} (hq : q.Monic) {x : S} (hx : (Polynomial.aeval x) q = 0) : (Polynomial.aeval x) (p.modByMonic q) = (Polynomial.aeval x) p

instance Polynomial.instNoZeroDivisors {R : Type u} [Semiring R] [NoZeroDivisors R] : NoZeroDivisors (Polynomial R)

Equations

• ⋯ = ⋯

theorem Polynomial.natDegree_mul {R : Type u} [Semiring R] [NoZeroDivisors R] {p : Polynomial R} {q : Polynomial R} (hp : p ≠ 0) (hq : q ≠ 0) : (p * q).natDegree = p.natDegree + q.natDegree

theorem Polynomial.trailingDegree_mul {R : Type u} [Semiring R] [NoZeroDivisors R] {p : Polynomial R} {q : Polynomial R} : (p * q).trailingDegree = p.trailingDegree + q.trailingDegree

@[simp]

theorem Polynomial.natDegree_pow {R : Type u} [Semiring R] [NoZeroDivisors R] (p : Polynomial R) (n : ℕ) : (p ^ n).natDegree = n * p.natDegree

theorem Polynomial.degree_le_mul_left {R : Type u} [Semiring R] [NoZeroDivisors R] {q : Polynomial R} (p : Polynomial R) (hq : q ≠ 0) : p.degree ≤ (p * q).degree

theorem Polynomial.natDegree_le_of_dvd {R : Type u} [Semiring R] [NoZeroDivisors R] {p : Polynomial R} {q : Polynomial R} (h1 : p ∣ q) (h2 : q ≠ 0) : p.natDegree ≤ q.natDegree

theorem Polynomial.degree_le_of_dvd {R : Type u} [Semiring R] [NoZeroDivisors R] {p : Polynomial R} {q : Polynomial R} (h1 : p ∣ q) (h2 : q ≠ 0) : p.degree ≤ q.degree

theorem Polynomial.eq_zero_of_dvd_of_degree_lt {R : Type u} [Semiring R] [NoZeroDivisors R] {p : Polynomial R} {q : Polynomial R} (h₁ : p ∣ q) (h₂ : q.degree < p.degree) : q = 0

theorem Polynomial.eq_zero_of_dvd_of_natDegree_lt {R : Type u} [Semiring R] [NoZeroDivisors R] {p : Polynomial R} {q : Polynomial R} (h₁ : p ∣ q) (h₂ : q.natDegree < p.natDegree) : q = 0

theorem Polynomial.not_dvd_of_degree_lt {R : Type u} [Semiring R] [NoZeroDivisors R] {p : Polynomial R} {q : Polynomial R} (h0 : q ≠ 0) (hl : q.degree < p.degree) : ¬p ∣ q

theorem Polynomial.not_dvd_of_natDegree_lt {R : Type u} [Semiring R] [NoZeroDivisors R] {p : Polynomial R} {q : Polynomial R} (h0 : q ≠ 0) (hl : q.natDegree < p.natDegree) : ¬p ∣ q

theorem Polynomial.natDegree_sub_eq_of_prod_eq {R : Type u} [Semiring R] [NoZeroDivisors R] {p₁ : Polynomial R} {p₂ : Polynomial R} {q₁ : Polynomial R} {q₂ : Polynomial R} (hp₁ : p₁ ≠ 0) (hq₁ : q₁ ≠ 0) (hp₂ : p₂ ≠ 0) (hq₂ : q₂ ≠ 0) (h_eq : p₁ * q₂ = p₂ * q₁) : ↑p₁.natDegree - ↑q₁.natDegree = ↑p₂.natDegree - ↑q₂.natDegree

This lemma is useful for working with the intDegree of a rational function.

theorem Polynomial.natDegree_eq_zero_of_isUnit {R : Type u} [Semiring R] [NoZeroDivisors R] {p : Polynomial R} (h : IsUnit p) : p.natDegree = 0

theorem Polynomial.degree_eq_zero_of_isUnit {R : Type u} [Semiring R] [NoZeroDivisors R] {p : Polynomial R} [Nontrivial R] (h : IsUnit p) : p.degree = 0

@[simp]

theorem Polynomial.degree_coe_units {R : Type u} [Semiring R] [NoZeroDivisors R] [Nontrivial R] (u : (Polynomial R)ˣ) : (↑u).degree = 0

theorem Polynomial.isUnit_iff {R : Type u} [Semiring R] [NoZeroDivisors R] {p : Polynomial R} : IsUnit p ↔ ∃ (r : R), IsUnit r ∧ Polynomial.C r = p

Characterization of a unit of a polynomial ring over an integral domain R. See Polynomial.isUnit_iff_coeff_isUnit_isNilpotent when R is a commutative ring.

theorem Polynomial.not_isUnit_of_degree_pos {R : Type u} [Semiring R] [NoZeroDivisors R] (p : Polynomial R) (hpl : 0 < p.degree) : ¬IsUnit p

theorem Polynomial.not_isUnit_of_natDegree_pos {R : Type u} [Semiring R] [NoZeroDivisors R] (p : Polynomial R) (hpl : 0 < p.natDegree) : ¬IsUnit p

theorem Polynomial.irreducible_of_monic {R : Type u} [CommSemiring R] [NoZeroDivisors R] {p : Polynomial R} (hp : p.Monic) (hp1 : p ≠ 1) : Irreducible p ↔ ∀ (f g : Polynomial R), f.Monic → g.Monic → f * g = p → f = 1 ∨ g = 1

theorem Polynomial.Monic.irreducible_iff_natDegree {R : Type u} [CommSemiring R] [NoZeroDivisors R] {p : Polynomial R} (hp : p.Monic) : Irreducible p ↔ p ≠ 1 ∧ ∀ (f g : Polynomial R), f.Monic → g.Monic → f * g = p → f.natDegree = 0 ∨ g.natDegree = 0

theorem Polynomial.Monic.irreducible_iff_natDegree' {R : Type u} [CommSemiring R] [NoZeroDivisors R] {p : Polynomial R} (hp : p.Monic) : Irreducible p ↔ p ≠ 1 ∧ ∀ (f g : Polynomial R), f.Monic → g.Monic → f * g = p → g.natDegree ∉ Finset.Ioc 0 (p.natDegree / 2)

theorem Polynomial.Monic.irreducible_iff_lt_natDegree_lt {R : Type u} [CommSemiring R] [NoZeroDivisors R] {p : Polynomial R} (hp : p.Monic) (hp1 : p ≠ 1) : Irreducible p ↔ ∀ (q : Polynomial R), q.Monic → q.natDegree ∈ Finset.Ioc 0 (p.natDegree / 2) → ¬q ∣ p

Alternate phrasing of Polynomial.Monic.irreducible_iff_natDegree' where we only have to check one divisor at a time.

theorem Polynomial.Monic.not_irreducible_iff_exists_add_mul_eq_coeff {R : Type u} [CommSemiring R] [NoZeroDivisors R] {p : Polynomial R} (hm : p.Monic) (hnd : p.natDegree = 2) : ¬Irreducible p ↔ ∃ (c₁ : R) (c₂ : R), p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂

theorem Polynomial.root_mul {R : Type u} {a : R} [CommSemiring R] [NoZeroDivisors R] {p : Polynomial R} {q : Polynomial R} : (p * q).IsRoot a ↔ p.IsRoot a ∨ q.IsRoot a

theorem Polynomial.root_or_root_of_root_mul {R : Type u} {a : R} [CommSemiring R] [NoZeroDivisors R] {p : Polynomial R} {q : Polynomial R} (h : (p * q).IsRoot a) : p.IsRoot a ∨ q.IsRoot a

instance Polynomial.instIsDomain {R : Type u} [Ring R] [IsDomain R] : IsDomain (Polynomial R)

Equations

• ⋯ = ⋯

theorem Polynomial.Monic.C_dvd_iff_isUnit {R : Type u} [CommSemiring R] {p : Polynomial R} (hp : p.Monic) {a : R} : Polynomial.C a ∣ p ↔ IsUnit a

theorem Polynomial.degree_pos_of_not_isUnit_of_dvd_monic {R : Type u} [CommSemiring R] {a : Polynomial R} {p : Polynomial R} (ha : ¬IsUnit a) (hap : a ∣ p) (hp : p.Monic) : 0 < a.degree

theorem Polynomial.natDegree_pos_of_not_isUnit_of_dvd_monic {R : Type u} [CommSemiring R] {a : Polynomial R} {p : Polynomial R} (ha : ¬IsUnit a) (hap : a ∣ p) (hp : p.Monic) : 0 < a.natDegree

theorem Polynomial.degree_pos_of_monic_of_not_isUnit {R : Type u} [CommSemiring R] {a : Polynomial R} (hu : ¬IsUnit a) (ha : a.Monic) : 0 < a.degree

theorem Polynomial.natDegree_pos_of_monic_of_not_isUnit {R : Type u} [CommSemiring R] {a : Polynomial R} (hu : ¬IsUnit a) (ha : a.Monic) : 0 < a.natDegree

theorem Polynomial.le_rootMultiplicity_iff {R : Type u} [CommRing R] {p : Polynomial R} (p0 : p ≠ 0) {a : R} {n : ℕ} : n ≤ Polynomial.rootMultiplicity a p ↔ (Polynomial.X - Polynomial.C a) ^ n ∣ p

The multiplicity of a as root of a nonzero polynomial p is at least n iff (X - a) ^ n divides p.

theorem Polynomial.rootMultiplicity_le_iff {R : Type u} [CommRing R] {p : Polynomial R} (p0 : p ≠ 0) (a : R) (n : ℕ) : Polynomial.rootMultiplicity a p ≤ n ↔ ¬(Polynomial.X - Polynomial.C a) ^ (n + 1) ∣ p

theorem Polynomial.pow_rootMultiplicity_not_dvd {R : Type u} [CommRing R] {p : Polynomial R} (p0 : p ≠ 0) (a : R) : ¬(Polynomial.X - Polynomial.C a) ^ (Polynomial.rootMultiplicity a p + 1) ∣ p

theorem Polynomial.X_sub_C_pow_dvd_iff {R : Type u} [CommRing R] {p : Polynomial R} {t : R} {n : ℕ} : (Polynomial.X - Polynomial.C t) ^ n ∣ p ↔ Polynomial.X ^ n ∣ p.comp (Polynomial.X + Polynomial.C t)

theorem Polynomial.comp_X_add_C_eq_zero_iff {R : Type u} [CommRing R] {p : Polynomial R} (t : R) : p.comp (Polynomial.X + Polynomial.C t) = 0 ↔ p = 0

theorem Polynomial.comp_X_add_C_ne_zero_iff {R : Type u} [CommRing R] {p : Polynomial R} (t : R) : p.comp (Polynomial.X + Polynomial.C t) ≠ 0 ↔ p ≠ 0

theorem Polynomial.rootMultiplicity_eq_rootMultiplicity {R : Type u} [CommRing R] {p : Polynomial R} {t : R} : Polynomial.rootMultiplicity t p = Polynomial.rootMultiplicity 0 (p.comp (Polynomial.X + Polynomial.C t))

theorem Polynomial.rootMultiplicity_eq_natTrailingDegree' {R : Type u} [CommRing R] {p : Polynomial R} : Polynomial.rootMultiplicity 0 p = p.natTrailingDegree

theorem Polynomial.rootMultiplicity_eq_natTrailingDegree {R : Type u} [CommRing R] {p : Polynomial R} {t : R} : Polynomial.rootMultiplicity t p = (p.comp (Polynomial.X + Polynomial.C t)).natTrailingDegree

theorem Polynomial.eval_divByMonic_eq_trailingCoeff_comp {R : Type u} [CommRing R] {p : Polynomial R} {t : R} : Polynomial.eval t (p.divByMonic ((Polynomial.X - Polynomial.C t) ^ Polynomial.rootMultiplicity t p)) = (p.comp (Polynomial.X + Polynomial.C t)).trailingCoeff

theorem Polynomial.Monic.mem_nonZeroDivisors {R : Type u} [CommRing R] {p : Polynomial R} (h : p.Monic) : p ∈ nonZeroDivisors (Polynomial R)

theorem Polynomial.mem_nonZeroDivisors_of_leadingCoeff {R : Type u} [CommRing R] {p : Polynomial R} (h : p.leadingCoeff ∈ nonZeroDivisors R) : p ∈ nonZeroDivisors (Polynomial R)

theorem Polynomial.rootMultiplicity_mul_X_sub_C_pow {R : Type u} [CommRing R] {p : Polynomial R} {a : R} {n : ℕ} (h : p ≠ 0) : Polynomial.rootMultiplicity a (p * (Polynomial.X - Polynomial.C a) ^ n) = Polynomial.rootMultiplicity a p + n

theorem Polynomial.rootMultiplicity_X_sub_C_pow {R : Type u} [CommRing R] [Nontrivial R] (a : R) (n : ℕ) : Polynomial.rootMultiplicity a ((Polynomial.X - Polynomial.C a) ^ n) = n

The multiplicity of a as root of (X - a) ^ n is n.

theorem Polynomial.rootMultiplicity_X_sub_C_self {R : Type u} [CommRing R] [Nontrivial R] {x : R} : Polynomial.rootMultiplicity x (Polynomial.X - Polynomial.C x) = 1

theorem Polynomial.rootMultiplicity_X_sub_C {R : Type u} [CommRing R] [Nontrivial R] [DecidableEq R] {x : R} {y : R} : Polynomial.rootMultiplicity x (Polynomial.X - Polynomial.C y) = if x = y then 1 else 0

theorem Polynomial.rootMultiplicity_add {R : Type u} [CommRing R] {p : Polynomial R} {q : Polynomial R} (a : R) (hzero : p + q ≠ 0) : min (Polynomial.rootMultiplicity a p) (Polynomial.rootMultiplicity a q) ≤ Polynomial.rootMultiplicity a (p + q)

The multiplicity of p + q is at least the minimum of the multiplicities.

theorem Polynomial.le_rootMultiplicity_mul {R : Type u} [CommRing R] {p : Polynomial R} {q : Polynomial R} (x : R) (hpq : p * q ≠ 0) : Polynomial.rootMultiplicity x p + Polynomial.rootMultiplicity x q ≤ Polynomial.rootMultiplicity x (p * q)

theorem Polynomial.rootMultiplicity_mul' {R : Type u} [CommRing R] {p : Polynomial R} {q : Polynomial R} {x : R} (hpq : Polynomial.eval x (p.divByMonic ((Polynomial.X - Polynomial.C x) ^ Polynomial.rootMultiplicity x p)) * Polynomial.eval x (q.divByMonic ((Polynomial.X - Polynomial.C x) ^ Polynomial.rootMultiplicity x q)) ≠ 0) : Polynomial.rootMultiplicity x (p * q) = Polynomial.rootMultiplicity x p + Polynomial.rootMultiplicity x q

theorem Polynomial.prime_X_sub_C {R : Type u} [CommRing R] [IsDomain R] (r : R) : Prime (Polynomial.X - Polynomial.C r)

theorem Polynomial.prime_X {R : Type u} [CommRing R] [IsDomain R] : Prime Polynomial.X

theorem Polynomial.Monic.prime_of_degree_eq_one {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hp1 : p.degree = 1) (hm : p.Monic) : Prime p

theorem Polynomial.irreducible_X_sub_C {R : Type u} [CommRing R] [IsDomain R] (r : R) : Irreducible (Polynomial.X - Polynomial.C r)

theorem Polynomial.irreducible_X {R : Type u} [CommRing R] [IsDomain R] : Irreducible Polynomial.X

theorem Polynomial.Monic.irreducible_of_degree_eq_one {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hp1 : p.degree = 1) (hm : p.Monic) : Irreducible p

theorem Polynomial.rootMultiplicity_mul {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {q : Polynomial R} {x : R} (hpq : p * q ≠ 0) : Polynomial.rootMultiplicity x (p * q) = Polynomial.rootMultiplicity x p + Polynomial.rootMultiplicity x q

theorem Polynomial.exists_multiset_roots {R : Type u} [CommRing R] [IsDomain R] [DecidableEq R] {p : Polynomial R} : p ≠ 0 → ∃ (s : Multiset R), ↑(Multiset.card s) ≤ p.degree ∧ ∀ (a : R), Multiset.count a s = Polynomial.rootMultiplicity a p

noncomputable def Polynomial.roots {R : Type u} [CommRing R] [IsDomain R] (p : Polynomial R) : Multiset R

roots p noncomputably gives a multiset containing all the roots of p, including their multiplicities.

Equations

• p.roots = if h : p = 0 then ∅ else Classical.choose ⋯

theorem Polynomial.roots_def {R : Type u} [CommRing R] [IsDomain R] [DecidableEq R] (p : Polynomial R) [Decidable (p = 0)] : p.roots = if h : p = 0 then ∅ else Classical.choose ⋯

@[simp]

theorem Polynomial.roots_zero {R : Type u} [CommRing R] [IsDomain R] : Polynomial.roots 0 = 0

theorem Polynomial.card_roots {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hp0 : p ≠ 0) : ↑(Multiset.card p.roots) ≤ p.degree

theorem Polynomial.card_roots' {R : Type u} [CommRing R] [IsDomain R] (p : Polynomial R) : Multiset.card p.roots ≤ p.natDegree

theorem Polynomial.card_roots_sub_C {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {a : R} (hp0 : 0 < p.degree) : ↑(Multiset.card (p - Polynomial.C a).roots) ≤ p.degree

theorem Polynomial.card_roots_sub_C' {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {a : R} (hp0 : 0 < p.degree) : Multiset.card (p - Polynomial.C a).roots ≤ p.natDegree

@[simp]

theorem Polynomial.count_roots {R : Type u} {a : R} [CommRing R] [IsDomain R] [DecidableEq R] (p : Polynomial R) : Multiset.count a p.roots = Polynomial.rootMultiplicity a p

@[simp]

theorem Polynomial.mem_roots' {R : Type u} {a : R} [CommRing R] [IsDomain R] {p : Polynomial R} : a ∈ p.roots ↔ p ≠ 0 ∧ p.IsRoot a

theorem Polynomial.mem_roots {R : Type u} {a : R} [CommRing R] [IsDomain R] {p : Polynomial R} (hp : p ≠ 0) : a ∈ p.roots ↔ p.IsRoot a

theorem Polynomial.ne_zero_of_mem_roots {R : Type u} {a : R} [CommRing R] [IsDomain R] {p : Polynomial R} (h : a ∈ p.roots) : p ≠ 0

theorem Polynomial.isRoot_of_mem_roots {R : Type u} {a : R} [CommRing R] [IsDomain R] {p : Polynomial R} (h : a ∈ p.roots) : p.IsRoot a

theorem Polynomial.mem_roots_iff_aeval_eq_zero {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {x : R} (w : p ≠ 0) : x ∈ p.roots ↔ (Polynomial.aeval x) p = 0

theorem Polynomial.card_le_degree_of_subset_roots {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {Z : Finset R} (h : Z.val ⊆ p.roots) : Z.card ≤ p.natDegree

theorem Polynomial.finite_setOf_isRoot {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hp : p ≠ 0) : {x : R | p.IsRoot x}.Finite

theorem Polynomial.eq_zero_of_infinite_isRoot {R : Type u} [CommRing R] [IsDomain R] (p : Polynomial R) (h : {x : R | p.IsRoot x}.Infinite) : p = 0

theorem Polynomial.exists_max_root {R : Type u} [CommRing R] [IsDomain R] [LinearOrder R] (p : Polynomial R) (hp : p ≠ 0) : ∃ (x₀ : R), ∀ (x : R), p.IsRoot x → x ≤ x₀

theorem Polynomial.exists_min_root {R : Type u} [CommRing R] [IsDomain R] [LinearOrder R] (p : Polynomial R) (hp : p ≠ 0) : ∃ (x₀ : R), ∀ (x : R), p.IsRoot x → x₀ ≤ x

theorem Polynomial.eq_of_infinite_eval_eq {R : Type u} [CommRing R] [IsDomain R] (p : Polynomial R) (q : Polynomial R) (h : {x : R | Polynomial.eval x p = Polynomial.eval x q}.Infinite) : p = q

theorem Polynomial.roots_mul {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {q : Polynomial R} (hpq : p * q ≠ 0) : (p * q).roots = p.roots + q.roots

theorem Polynomial.roots.le_of_dvd {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {q : Polynomial R} (h : q ≠ 0) : p ∣ q → p.roots ≤ q.roots

theorem Polynomial.mem_roots_sub_C' {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {a : R} {x : R} : x ∈ (p - Polynomial.C a).roots ↔ p ≠ Polynomial.C a ∧ Polynomial.eval x p = a

theorem Polynomial.mem_roots_sub_C {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {a : R} {x : R} (hp0 : 0 < p.degree) : x ∈ (p - Polynomial.C a).roots ↔ Polynomial.eval x p = a

@[simp]

theorem Polynomial.roots_X_sub_C {R : Type u} [CommRing R] [IsDomain R] (r : R) : (Polynomial.X - Polynomial.C r).roots = {r}

@[simp]

theorem Polynomial.roots_X {R : Type u} [CommRing R] [IsDomain R] : Polynomial.X.roots = {0}

@[simp]

theorem Polynomial.roots_C {R : Type u} [CommRing R] [IsDomain R] (x : R) : (Polynomial.C x).roots = 0

@[simp]

theorem Polynomial.roots_one {R : Type u} [CommRing R] [IsDomain R] : Polynomial.roots 1 = ∅

@[simp]

theorem Polynomial.roots_C_mul {R : Type u} {a : R} [CommRing R] [IsDomain R] (p : Polynomial R) (ha : a ≠ 0) : (Polynomial.C a * p).roots = p.roots

@[simp]

theorem Polynomial.roots_smul_nonzero {R : Type u} {a : R} [CommRing R] [IsDomain R] (p : Polynomial R) (ha : a ≠ 0) : (a • p).roots = p.roots

@[simp]

theorem Polynomial.roots_neg {R : Type u} [CommRing R] [IsDomain R] (p : Polynomial R) : (-p).roots = p.roots

theorem Polynomial.roots_list_prod {R : Type u} [CommRing R] [IsDomain R] (L : List (Polynomial R)) : 0 ∉ L → L.prod.roots = (↑L).bind Polynomial.roots

theorem Polynomial.roots_multiset_prod {R : Type u} [CommRing R] [IsDomain R] (m : Multiset (Polynomial R)) : 0 ∉ m → m.prod.roots = m.bind Polynomial.roots

theorem Polynomial.roots_prod {R : Type u} [CommRing R] [IsDomain R] {ι : Type u_1} (f : ι → Polynomial R) (s : Finset ι) : s.prod f ≠ 0 → (s.prod f).roots = s.val.bind fun (i : ι) => (f i).roots

@[simp]

theorem Polynomial.roots_pow {R : Type u} [CommRing R] [IsDomain R] (p : Polynomial R) (n : ℕ) : (p ^ n).roots = n • p.roots

theorem Polynomial.roots_X_pow {R : Type u} [CommRing R] [IsDomain R] (n : ℕ) : (Polynomial.X ^ n).roots = n • {0}

theorem Polynomial.roots_C_mul_X_pow {R : Type u} {a : R} [CommRing R] [IsDomain R] (ha : a ≠ 0) (n : ℕ) : (Polynomial.C a * Polynomial.X ^ n).roots = n • {0}

@[simp]

theorem Polynomial.roots_monomial {R : Type u} {a : R} [CommRing R] [IsDomain R] (ha : a ≠ 0) (n : ℕ) : ((Polynomial.monomial n) a).roots = n • {0}

theorem Polynomial.roots_prod_X_sub_C {R : Type u} [CommRing R] [IsDomain R] (s : Finset R) : (s.prod fun (a : R) => Polynomial.X - Polynomial.C a).roots = s.val

@[simp]

theorem Polynomial.roots_multiset_prod_X_sub_C {R : Type u} [CommRing R] [IsDomain R] (s : Multiset R) : (Multiset.map (fun (a : R) => Polynomial.X - Polynomial.C a) s).prod.roots = s

@[simp]

theorem Polynomial.natDegree_multiset_prod_X_sub_C_eq_card {R : Type u} [CommRing R] [IsDomain R] (s : Multiset R) : (Multiset.map (fun (a : R) => Polynomial.X - Polynomial.C a) s).prod.natDegree = Multiset.card s

theorem Polynomial.card_roots_X_pow_sub_C {R : Type u} [CommRing R] [IsDomain R] {n : ℕ} (hn : 0 < n) (a : R) : Multiset.card (Polynomial.X ^ n - Polynomial.C a).roots ≤ n

def Polynomial.nthRoots {R : Type u} [CommRing R] [IsDomain R] (n : ℕ) (a : R) : Multiset R

nthRoots n a noncomputably returns the solutions to x ^ n = a

Equations

• Polynomial.nthRoots n a = (Polynomial.X ^ n - Polynomial.C a).roots

@[simp]

theorem Polynomial.mem_nthRoots {R : Type u} [CommRing R] [IsDomain R] {n : ℕ} (hn : 0 < n) {a : R} {x : R} : x ∈ Polynomial.nthRoots n a ↔ x ^ n = a

@[simp]

theorem Polynomial.nthRoots_zero {R : Type u} [CommRing R] [IsDomain R] (r : R) : Polynomial.nthRoots 0 r = 0

@[simp]

theorem Polynomial.nthRoots_zero_right {R : Type u_1} [CommRing R] [IsDomain R] (n : ℕ) : Polynomial.nthRoots n 0 = Multiset.replicate n 0

theorem Polynomial.card_nthRoots {R : Type u} [CommRing R] [IsDomain R] (n : ℕ) (a : R) : Multiset.card (Polynomial.nthRoots n a) ≤ n

@[simp]

theorem Polynomial.nthRoots_two_eq_zero_iff {R : Type u} [CommRing R] [IsDomain R] {r : R} : Polynomial.nthRoots 2 r = 0 ↔ ¬IsSquare r

def Polynomial.nthRootsFinset (n : ℕ) (R : Type u_1) [CommRing R] [IsDomain R] : Finset R

The multiset nthRoots ↑n (1 : R) as a Finset.

Equations

• Polynomial.nthRootsFinset n R = (Polynomial.nthRoots n 1).toFinset

theorem Polynomial.nthRootsFinset_def (n : ℕ) (R : Type u_1) [CommRing R] [IsDomain R] [DecidableEq R] : Polynomial.nthRootsFinset n R = (Polynomial.nthRoots n 1).toFinset

@[simp]

theorem Polynomial.mem_nthRootsFinset {R : Type u} [CommRing R] [IsDomain R] {n : ℕ} (h : 0 < n) {x : R} : x ∈ Polynomial.nthRootsFinset n R ↔ x ^ n = 1

@[simp]

theorem Polynomial.nthRootsFinset_zero {R : Type u} [CommRing R] [IsDomain R] : Polynomial.nthRootsFinset 0 R = ∅

theorem Polynomial.mul_mem_nthRootsFinset {R : Type u} {n : ℕ} [CommRing R] [IsDomain R] {η₁ : R} {η₂ : R} (hη₁ : η₁ ∈ Polynomial.nthRootsFinset n R) (hη₂ : η₂ ∈ Polynomial.nthRootsFinset n R) : η₁ * η₂ ∈ Polynomial.nthRootsFinset n R

theorem Polynomial.ne_zero_of_mem_nthRootsFinset {R : Type u} {n : ℕ} [CommRing R] [IsDomain R] {η : R} (hη : η ∈ Polynomial.nthRootsFinset n R) : η ≠ 0

theorem Polynomial.one_mem_nthRootsFinset {R : Type u} {n : ℕ} [CommRing R] [IsDomain R] (hn : 0 < n) : 1 ∈ Polynomial.nthRootsFinset n R

theorem Polynomial.Monic.comp {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {q : Polynomial R} (hp : p.Monic) (hq : q.Monic) (h : q.natDegree ≠ 0) : (p.comp q).Monic

theorem Polynomial.Monic.comp_X_add_C {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hp : p.Monic) (r : R) : (p.comp (Polynomial.X + Polynomial.C r)).Monic

theorem Polynomial.Monic.comp_X_sub_C {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hp : p.Monic) (r : R) : (p.comp (Polynomial.X - Polynomial.C r)).Monic

theorem Polynomial.units_coeff_zero_smul {R : Type u} [CommRing R] [IsDomain R] (c : (Polynomial R)ˣ) (p : Polynomial R) : (↑c).coeff 0 • p = ↑c * p

@[simp]

theorem Polynomial.natDegree_coe_units {R : Type u} [CommRing R] [IsDomain R] (u : (Polynomial R)ˣ) : (↑u).natDegree = 0

theorem Polynomial.comp_eq_zero_iff {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {q : Polynomial R} : p.comp q = 0 ↔ p = 0 ∨ Polynomial.eval (q.coeff 0) p = 0 ∧ q = Polynomial.C (q.coeff 0)

theorem Polynomial.zero_of_eval_zero {R : Type u} [CommRing R] [IsDomain R] [Infinite R] (p : Polynomial R) (h : ∀ (x : R), Polynomial.eval x p = 0) : p = 0

theorem Polynomial.funext {R : Type u} [CommRing R] [IsDomain R] [Infinite R] {p : Polynomial R} {q : Polynomial R} (ext : ∀ (r : R), Polynomial.eval r p = Polynomial.eval r q) : p = q

@[reducible, inline]

noncomputable abbrev Polynomial.aroots {T : Type w} [CommRing T] (p : Polynomial T) (S : Type u_1) [CommRing S] [IsDomain S] [Algebra T S] : Multiset S

Given a polynomial p with coefficients in a ring T and a T-algebra S, aroots p S is the multiset of roots of p regarded as a polynomial over S.

Equations

• p.aroots S = (Polynomial.map (algebraMap T S) p).roots

theorem Polynomial.aroots_def {T : Type w} [CommRing T] (p : Polynomial T) (S : Type u_1) [CommRing S] [IsDomain S] [Algebra T S] : p.aroots S = (Polynomial.map (algebraMap T S) p).roots

theorem Polynomial.mem_aroots' {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] {p : Polynomial T} {a : S} : a ∈ p.aroots S ↔ Polynomial.map (algebraMap T S) p ≠ 0 ∧ (Polynomial.aeval a) p = 0

theorem Polynomial.mem_aroots {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] [NoZeroSMulDivisors T S] {p : Polynomial T} {a : S} : a ∈ p.aroots S ↔ p ≠ 0 ∧ (Polynomial.aeval a) p = 0

theorem Polynomial.aroots_mul {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] [NoZeroSMulDivisors T S] {p : Polynomial T} {q : Polynomial T} (hpq : p * q ≠ 0) : (p * q).aroots S = p.aroots S + q.aroots S

@[simp]

theorem Polynomial.aroots_X_sub_C {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] (r : T) : (Polynomial.X - Polynomial.C r).aroots S = {(algebraMap T S) r}

@[simp]

theorem Polynomial.aroots_X {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] : Polynomial.X.aroots S = {0}

@[simp]

theorem Polynomial.aroots_C {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] (a : T) : (Polynomial.C a).aroots S = 0

@[simp]

theorem Polynomial.aroots_zero {T : Type w} [CommRing T] (S : Type u_1) [CommRing S] [IsDomain S] [Algebra T S] : Polynomial.aroots 0 S = 0

@[simp]

theorem Polynomial.aroots_one {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] : Polynomial.aroots 1 S = 0

@[simp]

theorem Polynomial.aroots_neg {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] (p : Polynomial T) : (-p).aroots S = p.aroots S

@[simp]

theorem Polynomial.aroots_C_mul {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] [NoZeroSMulDivisors T S] {a : T} (p : Polynomial T) (ha : a ≠ 0) : (Polynomial.C a * p).aroots S = p.aroots S

@[simp]

theorem Polynomial.aroots_smul_nonzero {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] [NoZeroSMulDivisors T S] {a : T} (p : Polynomial T) (ha : a ≠ 0) : (a • p).aroots S = p.aroots S

@[simp]

theorem Polynomial.aroots_pow {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] (p : Polynomial T) (n : ℕ) : (p ^ n).aroots S = n • p.aroots S

theorem Polynomial.aroots_X_pow {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] (n : ℕ) : (Polynomial.X ^ n).aroots S = n • {0}

theorem Polynomial.aroots_C_mul_X_pow {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] [NoZeroSMulDivisors T S] {a : T} (ha : a ≠ 0) (n : ℕ) : (Polynomial.C a * Polynomial.X ^ n).aroots S = n • {0}

@[simp]

theorem Polynomial.aroots_monomial {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] [NoZeroSMulDivisors T S] {a : T} (ha : a ≠ 0) (n : ℕ) : ((Polynomial.monomial n) a).aroots S = n • {0}

def Polynomial.rootSet {T : Type w} [CommRing T] (p : Polynomial T) (S : Type u_1) [CommRing S] [IsDomain S] [Algebra T S] : Set S

The set of distinct roots of p in S.

If you have a non-separable polynomial, use Polynomial.aroots for the multiset where multiple roots have the appropriate multiplicity.

Equations

• p.rootSet S = ↑(p.aroots S).toFinset

theorem Polynomial.rootSet_def {T : Type w} [CommRing T] (p : Polynomial T) (S : Type u_1) [CommRing S] [IsDomain S] [Algebra T S] [DecidableEq S] : p.rootSet S = ↑(p.aroots S).toFinset

@[simp]

theorem Polynomial.rootSet_C {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] (a : T) : (Polynomial.C a).rootSet S = ∅

@[simp]

theorem Polynomial.rootSet_zero {T : Type w} [CommRing T] (S : Type u_1) [CommRing S] [IsDomain S] [Algebra T S] : Polynomial.rootSet 0 S = ∅

@[simp]

theorem Polynomial.rootSet_one {T : Type w} [CommRing T] (S : Type u_1) [CommRing S] [IsDomain S] [Algebra T S] : Polynomial.rootSet 1 S = ∅

@[simp]

theorem Polynomial.rootSet_neg {T : Type w} [CommRing T] (p : Polynomial T) (S : Type u_1) [CommRing S] [IsDomain S] [Algebra T S] : (-p).rootSet S = p.rootSet S

instance Polynomial.rootSetFintype {T : Type w} [CommRing T] (p : Polynomial T) (S : Type u_1) [CommRing S] [IsDomain S] [Algebra T S] : Fintype ↑(p.rootSet S)

Equations

• p.rootSetFintype S = FinsetCoe.fintype (p.aroots S).toFinset

theorem Polynomial.rootSet_finite {T : Type w} [CommRing T] (p : Polynomial T) (S : Type u_1) [CommRing S] [IsDomain S] [Algebra T S] : (p.rootSet S).Finite

theorem Polynomial.bUnion_roots_finite {R : Type u_1} {S : Type u_2} [Semiring R] [CommRing S] [IsDomain S] [DecidableEq S] (m : R →+* S) (d : ℕ) {U : Set R} (h : U.Finite) : (⋃ (f : Polynomial R), ⋃ (_ : f.natDegree ≤ d ∧ ∀ (i : ℕ), f.coeff i ∈ U), ↑(Polynomial.map m f).roots.toFinset).Finite

The set of roots of all polynomials of bounded degree and having coefficients in a finite set is finite.

theorem Polynomial.mem_rootSet' {T : Type w} [CommRing T] {p : Polynomial T} {S : Type u_1} [CommRing S] [IsDomain S] [Algebra T S] {a : S} : a ∈ p.rootSet S ↔ Polynomial.map (algebraMap T S) p ≠ 0 ∧ (Polynomial.aeval a) p = 0

theorem Polynomial.mem_rootSet {T : Type w} [CommRing T] {p : Polynomial T} {S : Type u_1} [CommRing S] [IsDomain S] [Algebra T S] [NoZeroSMulDivisors T S] {a : S} : a ∈ p.rootSet S ↔ p ≠ 0 ∧ (Polynomial.aeval a) p = 0

theorem Polynomial.mem_rootSet_of_ne {T : Type w} [CommRing T] {p : Polynomial T} {S : Type u_1} [CommRing S] [IsDomain S] [Algebra T S] [NoZeroSMulDivisors T S] (hp : p ≠ 0) {a : S} : a ∈ p.rootSet S ↔ (Polynomial.aeval a) p = 0

theorem Polynomial.rootSet_maps_to' {T : Type w} [CommRing T] {p : Polynomial T} {S : Type u_1} {S' : Type u_2} [CommRing S] [IsDomain S] [Algebra T S] [CommRing S'] [IsDomain S'] [Algebra T S'] (hp : Polynomial.map (algebraMap T S') p = 0 → Polynomial.map (algebraMap T S) p = 0) (f : S →ₐ[T] S') : Set.MapsTo (⇑f) (p.rootSet S) (p.rootSet S')

theorem Polynomial.ne_zero_of_mem_rootSet {S : Type v} {T : Type w} [CommRing T] {p : Polynomial T} [CommRing S] [IsDomain S] [Algebra T S] {a : S} (h : a ∈ p.rootSet S) : p ≠ 0

theorem Polynomial.aeval_eq_zero_of_mem_rootSet {S : Type v} {T : Type w} [CommRing T] {p : Polynomial T} [CommRing S] [IsDomain S] [Algebra T S] {a : S} (hx : a ∈ p.rootSet S) : (Polynomial.aeval a) p = 0

theorem Polynomial.rootSet_mapsTo {T : Type w} [CommRing T] {p : Polynomial T} {S : Type u_1} {S' : Type u_2} [CommRing S] [IsDomain S] [Algebra T S] [CommRing S'] [IsDomain S'] [Algebra T S'] [NoZeroSMulDivisors T S'] (f : S →ₐ[T] S') : Set.MapsTo (⇑f) (p.rootSet S) (p.rootSet S')

theorem Polynomial.coeff_coe_units_zero_ne_zero {R : Type u} [CommRing R] [IsDomain R] (u : (Polynomial R)ˣ) : (↑u).coeff 0 ≠ 0

theorem Polynomial.degree_eq_degree_of_associated {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {q : Polynomial R} (h : Associated p q) : p.degree = q.degree

theorem Polynomial.degree_eq_one_of_irreducible_of_root {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hi : Irreducible p) {x : R} (hx : p.IsRoot x) : p.degree = 1

theorem Polynomial.leadingCoeff_divByMonic_of_monic {R : Type u} [CommRing R] {p : Polynomial R} {q : Polynomial R} (hmonic : q.Monic) (hdegree : q.degree ≤ p.degree) : (p.divByMonic q).leadingCoeff = p.leadingCoeff

Division by a monic polynomial doesn't change the leading coefficient.

theorem Polynomial.leadingCoeff_divByMonic_X_sub_C {R : Type u} [CommRing R] [IsDomain R] (p : Polynomial R) (hp : p.degree ≠ 0) (a : R) : (p.divByMonic (Polynomial.X - Polynomial.C a)).leadingCoeff = p.leadingCoeff

theorem Polynomial.eq_of_dvd_of_natDegree_le_of_leadingCoeff {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {q : Polynomial R} (hpq : p ∣ q) (h₁ : q.natDegree ≤ p.natDegree) (h₂ : p.leadingCoeff = q.leadingCoeff) : p = q

theorem Polynomial.associated_of_dvd_of_natDegree_le_of_leadingCoeff {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {q : Polynomial R} (hpq : p ∣ q) (h₁ : q.natDegree ≤ p.natDegree) (h₂ : q.leadingCoeff ∣ p.leadingCoeff) : Associated p q

theorem Polynomial.associated_of_dvd_of_natDegree_le {K : Type u_1} [Field K] {p : Polynomial K} {q : Polynomial K} (hpq : p ∣ q) (hq : q ≠ 0) (h₁ : q.natDegree ≤ p.natDegree) : Associated p q

theorem Polynomial.associated_of_dvd_of_degree_eq {K : Type u_1} [Field K] {p : Polynomial K} {q : Polynomial K} (hpq : p ∣ q) (h₁ : p.degree = q.degree) : Associated p q

theorem Polynomial.eq_leadingCoeff_mul_of_monic_of_dvd_of_natDegree_le {R : Type u_1} [CommRing R] {p : Polynomial R} {q : Polynomial R} (hp : p.Monic) (hdiv : p ∣ q) (hdeg : q.natDegree ≤ p.natDegree) : q = Polynomial.C q.leadingCoeff * p

theorem Polynomial.eq_of_monic_of_dvd_of_natDegree_le {R : Type u_1} [CommRing R] {p : Polynomial R} {q : Polynomial R} (hp : p.Monic) (hq : q.Monic) (hdiv : p ∣ q) (hdeg : q.natDegree ≤ p.natDegree) : q = p

theorem Polynomial.eq_zero_of_natDegree_lt_card_of_eval_eq_zero {R : Type u_1} [CommRing R] [IsDomain R] (p : Polynomial R) {ι : Type u_2} [Fintype ι] {f : ι → R} (hf : Function.Injective f) (heval : ∀ (i : ι), Polynomial.eval (f i) p = 0) (hcard : p.natDegree < Fintype.card ι) : p = 0

theorem Polynomial.eq_zero_of_natDegree_lt_card_of_eval_eq_zero' {R : Type u_1} [CommRing R] [IsDomain R] (p : Polynomial R) (s : Finset R) (heval : ∀ i ∈ s, Polynomial.eval i p = 0) (hcard : p.natDegree < s.card) : p = 0

theorem Polynomial.eq_zero_of_forall_eval_zero_of_natDegree_lt_card {R : Type u} [CommRing R] [IsDomain R] (f : Polynomial R) (hf : ∀ (r : R), Polynomial.eval r f = 0) (hfR : ↑f.natDegree < Cardinal.mk R) : f = 0

theorem Polynomial.exists_eval_ne_zero_of_natDegree_lt_card {R : Type u} [CommRing R] [IsDomain R] (f : Polynomial R) (hf : f ≠ 0) (hfR : ↑f.natDegree < Cardinal.mk R) : ∃ (r : R), Polynomial.eval r f ≠ 0

theorem Polynomial.isCoprime_X_sub_C_of_isUnit_sub {R : Type u_1} [CommRing R] {a : R} {b : R} (h : IsUnit (a - b)) : IsCoprime (Polynomial.X - Polynomial.C a) (Polynomial.X - Polynomial.C b)

theorem Polynomial.pairwise_coprime_X_sub_C {K : Type u_1} [Field K] {I : Type v} {s : I → K} (H : Function.Injective s) : Pairwise (IsCoprime on fun (i : I) => Polynomial.X - Polynomial.C (s i))

theorem Polynomial.monic_prod_multiset_X_sub_C {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} : (Multiset.map (fun (a : R) => Polynomial.X - Polynomial.C a) p.roots).prod.Monic

theorem Polynomial.prod_multiset_root_eq_finset_root {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} [DecidableEq R] : (Multiset.map (fun (a : R) => Polynomial.X - Polynomial.C a) p.roots).prod = p.roots.toFinset.prod fun (a : R) => (Polynomial.X - Polynomial.C a) ^ Polynomial.rootMultiplicity a p

theorem Polynomial.prod_multiset_X_sub_C_dvd {R : Type u} [CommRing R] [IsDomain R] (p : Polynomial R) : (Multiset.map (fun (a : R) => Polynomial.X - Polynomial.C a) p.roots).prod ∣ p

The product ∏ (X - a) for a inside the multiset p.roots divides p.

theorem Multiset.prod_X_sub_C_dvd_iff_le_roots {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hp : p ≠ 0) (s : Multiset R) : (Multiset.map (fun (a : R) => Polynomial.X - Polynomial.C a) s).prod ∣ p ↔ s ≤ p.roots

A Galois connection.

theorem Polynomial.exists_prod_multiset_X_sub_C_mul {R : Type u} [CommRing R] [IsDomain R] (p : Polynomial R) : ∃ (q : Polynomial R), (Multiset.map (fun (a : R) => Polynomial.X - Polynomial.C a) p.roots).prod * q = p ∧ Multiset.card p.roots + q.natDegree = p.natDegree ∧ q.roots = 0

theorem Polynomial.C_leadingCoeff_mul_prod_multiset_X_sub_C {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hroots : Multiset.card p.roots = p.natDegree) : Polynomial.C p.leadingCoeff * (Multiset.map (fun (a : R) => Polynomial.X - Polynomial.C a) p.roots).prod = p

A polynomial p that has as many roots as its degree can be written p = p.leadingCoeff * ∏(X - a), for a in p.roots.

theorem Polynomial.prod_multiset_X_sub_C_of_monic_of_roots_card_eq {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hp : p.Monic) (hroots : Multiset.card p.roots = p.natDegree) : (Multiset.map (fun (a : R) => Polynomial.X - Polynomial.C a) p.roots).prod = p

A monic polynomial p that has as many roots as its degree can be written p = ∏(X - a), for a in p.roots.

theorem Polynomial.Monic.isUnit_leadingCoeff_of_dvd {R : Type u} [CommRing R] [IsDomain R] {a : Polynomial R} {p : Polynomial R} (hp : p.Monic) (hap : a ∣ p) : IsUnit a.leadingCoeff

theorem Polynomial.Monic.irreducible_iff_degree_lt {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (p_monic : p.Monic) (p_1 : p ≠ 1) : Irreducible p ↔ ∀ (q : Polynomial R), q.degree ≤ ↑(p.natDegree / 2) → q ∣ p → IsUnit q

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

See also: Polynomial.Monic.irreducible_iff_natDegree.

theorem Polynomial.le_rootMultiplicity_map {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] {p : Polynomial A} {f : A →+* B} (hmap : Polynomial.map f p ≠ 0) (a : A) : Polynomial.rootMultiplicity a p ≤ Polynomial.rootMultiplicity (f a) (Polynomial.map f p)

theorem Polynomial.eq_rootMultiplicity_map {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] {p : Polynomial A} {f : A →+* B} (hf : Function.Injective ⇑f) (a : A) : Polynomial.rootMultiplicity a p = Polynomial.rootMultiplicity (f a) (Polynomial.map f p)

theorem Polynomial.count_map_roots {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [IsDomain A] [DecidableEq B] {p : Polynomial A} {f : A →+* B} (hmap : Polynomial.map f p ≠ 0) (b : B) : Multiset.count b (Multiset.map (⇑f) p.roots) ≤ Polynomial.rootMultiplicity b (Polynomial.map f p)

theorem Polynomial.count_map_roots_of_injective {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [IsDomain A] [DecidableEq B] (p : Polynomial A) {f : A →+* B} (hf : Function.Injective ⇑f) (b : B) : Multiset.count b (Multiset.map (⇑f) p.roots) ≤ Polynomial.rootMultiplicity b (Polynomial.map f p)

theorem Polynomial.map_roots_le {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [IsDomain A] [IsDomain B] {p : Polynomial A} {f : A →+* B} (h : Polynomial.map f p ≠ 0) : Multiset.map (⇑f) p.roots ≤ (Polynomial.map f p).roots

theorem Polynomial.map_roots_le_of_injective {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [IsDomain A] [IsDomain B] (p : Polynomial A) {f : A →+* B} (hf : Function.Injective ⇑f) : Multiset.map (⇑f) p.roots ≤ (Polynomial.map f p).roots

theorem Polynomial.card_roots_le_map {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [IsDomain A] [IsDomain B] {p : Polynomial A} {f : A →+* B} (h : Polynomial.map f p ≠ 0) : Multiset.card p.roots ≤ Multiset.card (Polynomial.map f p).roots

theorem Polynomial.card_roots_le_map_of_injective {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [IsDomain A] [IsDomain B] {p : Polynomial A} {f : A →+* B} (hf : Function.Injective ⇑f) : Multiset.card p.roots ≤ Multiset.card (Polynomial.map f p).roots

theorem Polynomial.roots_map_of_injective_of_card_eq_natDegree {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [IsDomain A] [IsDomain B] {p : Polynomial A} {f : A →+* B} (hf : Function.Injective ⇑f) (hroots : Multiset.card p.roots = p.natDegree) : Multiset.map (⇑f) p.roots = (Polynomial.map f p).roots

theorem Polynomial.isUnit_of_isUnit_leadingCoeff_of_isUnit_map {R : Type u} {S : Type v} [Semiring R] [CommRing S] [IsDomain S] (φ : R →+* S) {f : Polynomial R} (hf : IsUnit f.leadingCoeff) (H : IsUnit (Polynomial.map φ f)) : IsUnit f

theorem Polynomial.Monic.irreducible_of_irreducible_map {R : Type u} {S : Type v} [CommRing R] [IsDomain R] [CommRing S] [IsDomain S] (φ : R →+* S) (f : Polynomial R) (h_mon : f.Monic) (h_irr : Irreducible (Polynomial.map φ f)) : Irreducible f

A polynomial over an integral domain R is irreducible if it is monic and irreducible after mapping into an integral domain S.

A special case of this lemma is that a polynomial over ℤ is irreducible if it is monic and irreducible over ℤ/pℤ for some prime p.