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 : derivative p ≠ 0) : rootMultiplicity t p - 1 ≤ rootMultiplicity t (derivative p)

theorem Polynomial.derivative_rootMultiplicity_of_root_of_mem_nonZeroDivisors {R : Type u} [CommRing R] {p : Polynomial R} {t : R} (hpt : p.IsRoot t) (hnzd : ↑(rootMultiplicity t p) ∈ nonZeroDivisors R) : rootMultiplicity t (derivative p) = rootMultiplicity t p - 1

theorem Polynomial.isRoot_iterate_derivative_of_lt_rootMultiplicity {R : Type u} [CommRing R] {p : Polynomial R} {t : R} {n : ℕ} (hn : n < rootMultiplicity t p) : ((⇑derivative)^[n] p).IsRoot t

theorem Polynomial.eval_iterate_derivative_rootMultiplicity {R : Type u} [CommRing R] {p : Polynomial R} {t : R} : eval t ((⇑derivative)^[rootMultiplicity t p] p) = (rootMultiplicity t p).factorial • eval t (p /ₘ (X - C t) ^ 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 : ℕ} (h : p ≠ 0) (hroot : ∀ m ≤ n, ((⇑derivative)^[m] p).IsRoot t) (hnzd : ↑n.factorial ∈ nonZeroDivisors R) : n < 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 : ℕ} (h : p ≠ 0) (hroot : ∀ m ≤ n, ((⇑derivative)^[m] p).IsRoot t) (hnzd : ∀ m ≤ n, m ≠ 0 → ↑m ∈ nonZeroDivisors R) : n < rootMultiplicity t p

theorem Polynomial.lt_rootMultiplicity_iff_isRoot_iterate_derivative_of_mem_nonZeroDivisors {R : Type u} [CommRing R] {p : Polynomial R} {t : R} {n : ℕ} (h : p ≠ 0) (hnzd : ↑n.factorial ∈ nonZeroDivisors R) : n < rootMultiplicity t p ↔ ∀ m ≤ n, ((⇑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 : ℕ} (h : p ≠ 0) (hnzd : ∀ m ≤ n, m ≠ 0 → ↑m ∈ nonZeroDivisors R) : n < rootMultiplicity t p ↔ ∀ m ≤ n, ((⇑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 < rootMultiplicity t p ↔ ∀ m ≤ 1, ((⇑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 < rootMultiplicity t p ↔ p.IsRoot t ∧ (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 < rootMultiplicity t p ↔ (gcd p (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) : rootMultiplicity t (derivative p) = rootMultiplicity t p - 1

theorem Polynomial.rootMultiplicity_sub_one_le_derivative_rootMultiplicity {R : Type u} [CommRing R] [IsDomain R] [CharZero R] (p : Polynomial R) (t : R) : rootMultiplicity t p - 1 ≤ rootMultiplicity t (derivative p)

theorem Polynomial.lt_rootMultiplicity_of_isRoot_iterate_derivative {R : Type u} [CommRing R] [IsDomain R] [CharZero R] {p : Polynomial R} {t : R} {n : ℕ} (h : p ≠ 0) (hroot : ∀ m ≤ n, ((⇑derivative)^[m] p).IsRoot t) : n < rootMultiplicity t p

theorem Polynomial.lt_rootMultiplicity_iff_isRoot_iterate_derivative {R : Type u} [CommRing R] [IsDomain R] [CharZero R] {p : Polynomial R} {t : R} {n : ℕ} (h : p ≠ 0) : n < rootMultiplicity t p ↔ ∀ m ≤ n, ((⇑derivative)^[m] p).IsRoot t

theorem Polynomial.isRoot_of_isRoot_of_dvd_derivative_mul {R : Type u} [CommRing R] [IsDomain R] [CharZero R] {f g : Polynomial R} (hf0 : f ≠ 0) (hfd : f ∣ derivative f * g) {a : R} (haf : f.IsRoot a) : g.IsRoot a

A sufficient condition for the set of roots of a nonzero polynomial f to be a subset of the set of roots of g is that f divides f.derivative * g. Over an algebraically closed field of characteristic zero, this is also a necessary condition. See isRoot_of_isRoot_iff_dvd_derivative_mul

instance Polynomial.instNormalizationMonoid {R : Type u} [CommRing R] [IsDomain R] [NormalizationMonoid R] : NormalizationMonoid (Polynomial R)

@[simp]

theorem Polynomial.coe_normUnit {R : Type u} [CommRing R] [IsDomain R] [NormalizationMonoid R] {p : Polynomial R} : ↑(normUnit p) = C ↑(normUnit p.leadingCoeff)

@[simp]

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 X = 1

theorem Polynomial.X_eq_normalize {R : Type u} [CommRing R] [IsDomain R] [NormalizationMonoid R] : X = normalize 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) : 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) : map f 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) : (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) : (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) : (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} : (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 q : Polynomial R) : Polynomial R

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

def Polynomial.mod {R : Type u} [Field R] (p q : Polynomial R) : Polynomial R

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

instance Polynomial.instDiv {R : Type u} [Field R] : Div (Polynomial R)

instance Polynomial.instMod {R : Type u} [Field R] : Mod (Polynomial R)

theorem Polynomial.div_def {R : Type u} [Field R] {p q : Polynomial R} : p / q = C q.leadingCoeff⁻¹ * (p /ₘ (q * C q.leadingCoeff⁻¹))

theorem Polynomial.mod_def {R : Type u} [Field R] {p q : Polynomial R} : p % q = p %ₘ (q * 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 % (X - C a) = C (eval a p)

theorem Polynomial.mul_div_eq_iff_isRoot {R : Type u} {a : R} [Field R] {p : Polynomial R} : (X - C a) * (p / (X - C a)) = p ↔ p.IsRoot a

instance Polynomial.instEuclideanDomain {R : Type u} [Field R] : EuclideanDomain (Polynomial R)

theorem Polynomial.mod_eq_self_iff {R : Type u} [Field R] {p 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 q : Polynomial R} (hq0 : q ≠ 0) : p / q = 0 ↔ p.degree < q.degree

theorem Polynomial.degree_add_div {R : Type u} [Field R] {p 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 q : Polynomial R) : (p / q).degree ≤ p.degree

theorem Polynomial.degree_div_lt {R : Type u} [Field R] {p 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) : IsUnit (map f p) ↔ IsUnit p

theorem Polynomial.map_div {R : Type u} {k : Type y} [Field R] {p q : Polynomial R} [Field k] (f : R →+* k) : map f (p / q) = map f p / map f q

theorem Polynomial.map_mod {R : Type u} {k : Type y} [Field R] {p q : Polynomial R} [Field k] (f : R →+* k) : map f (p % q) = map f p % map f q

theorem Polynomial.natDegree_mod_lt {k : Type y} [Field k] (p : Polynomial k) {q : Polynomial k} (hq : q.natDegree ≠ 0) : (p % q).natDegree < q.natDegree

theorem Polynomial.gcd_map {R : Type u} {k : Type y} [Field R] {p q : Polynomial R} [Field k] [DecidableEq R] [DecidableEq k] (f : R →+* k) : EuclideanDomain.gcd (map f p) (map f q) = map f (EuclideanDomain.gcd p q)

theorem Polynomial.eval₂_gcd_eq_zero {R : Type u} {k : Type y} [Field R] [CommSemiring k] [DecidableEq R] {ϕ : R →+* k} {f g : Polynomial R} {α : k} (hf : eval₂ ϕ α f = 0) (hg : eval₂ ϕ α g = 0) : eval₂ ϕ α (EuclideanDomain.gcd f g) = 0

theorem Polynomial.eval_gcd_eq_zero {R : Type u} [Field R] [DecidableEq R] {f g : Polynomial R} {α : R} (hf : eval α f = 0) (hg : eval α g = 0) : eval α (EuclideanDomain.gcd f 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 g : Polynomial R} {α : k} (hα : eval₂ ϕ α (EuclideanDomain.gcd f g) = 0) : eval₂ ϕ α f = 0

theorem Polynomial.root_right_of_root_gcd {R : Type u} {k : Type y} [Field R] [CommSemiring k] [DecidableEq R] {ϕ : R →+* k} {f g : Polynomial R} {α : k} (hα : eval₂ ϕ α (EuclideanDomain.gcd f g) = 0) : eval₂ ϕ α g = 0

theorem Polynomial.root_gcd_iff_root_left_right {R : Type u} {k : Type y} [Field R] [CommSemiring k] [DecidableEq R] {ϕ : R →+* k} {f g : Polynomial R} {α : k} : eval₂ ϕ α (EuclideanDomain.gcd f g) = 0 ↔ eval₂ ϕ α f = 0 ∧ eval₂ ϕ α g = 0

theorem Polynomial.isRoot_gcd_iff_isRoot_left_right {R : Type u} [Field R] [DecidableEq R] {f 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 q : Polynomial R} [Field k] (f : R →+* k) : IsCoprime (map f p) (map f q) ↔ IsCoprime p q

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 ∈ (map f p).roots ↔ 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 : ℕ} (hn : n ≠ 0) {a : R} (ha : a ≠ 0) : ((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 : ℕ} (hn : n ≠ 0) {a : R} (ha : a ≠ 0) : (C a * 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 : ℕ} (hn : n ≠ 0) : (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 : ι → Polynomial R) (s : Finset ι) (h : s.prod f ≠ 0) : (s.prod f).rootSet S = ⋃ i ∈ s, (f i).rootSet S

theorem Polynomial.roots_C_mul_X_sub_C {R : Type u} {a : R} [Field R] (b : R) (ha : a ≠ 0) : (C a * X - C b).roots = {a⁻¹ * b}

theorem Polynomial.roots_C_mul_X_add_C {R : Type u} {a : R} [Field R] (b : R) (ha : a ≠ 0) : (C a * X + C b).roots = {-(a⁻¹ * b)}

theorem Polynomial.roots_degree_eq_one {R : Type u} [Field R] {p : Polynomial R} (h : p.degree = 1) : p.roots = {-((p.coeff 1)⁻¹ * p.coeff 0)}

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 : 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.normalize_eq_self_iff_monic {R : Type u} [Field R] [DecidableEq R] {p : Polynomial R} (hp : p ≠ 0) : normalize p = p ↔ p.Monic

theorem Polynomial.leadingCoeff_div {R : Type u} [Field R] {p 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 q : Polynomial R} : p / (C a * q) = C a⁻¹ * (p / q)

theorem Polynomial.div_C {R : Type u} {a : R} [Field R] {p : Polynomial R} : p / C a = p * C a⁻¹

theorem Polynomial.C_div {R : Type u} {a b : R} [Field R] : C (a / b) = C a / C b

theorem Polynomial.C_mul_dvd {R : Type u} {a : R} [Field R] {p q : Polynomial R} (ha : a ≠ 0) : C a * p ∣ q ↔ p ∣ q

theorem Polynomial.dvd_C_mul {R : Type u} {a : R} [Field R] {p q : Polynomial R} (ha : a ≠ 0) : p ∣ 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) = C p.leadingCoeff⁻¹

theorem Polynomial.map_dvd_map' {R : Type u} {k : Type y} [Field R] [Field k] (f : R →+* k) {x y : Polynomial R} : map f x ∣ map f y ↔ x ∣ y

@[simp]

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) : Prime p

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

theorem Polynomial.not_irreducible_C {R : Type u} [Field R] (x : R) : ¬Irreducible (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} [Ring K] (f : Polynomial K) (a : K) : (X - C a) * (f /ₘ (X - C a)) = f - f %ₘ (X - C a)

theorem Polynomial.divByMonic_add_X_sub_C_mul_derivative_divByMonic_eq_derivative {K : Type u_1} [CommRing K] (f : Polynomial K) (a : K) : f /ₘ (X - C a) + (X - C a) * derivative (f /ₘ (X - C a)) = derivative f

@[deprecated Polynomial.divByMonic_add_X_sub_C_mul_derivative_divByMonic_eq_derivative (since := "2025-07-08")]

theorem Polynomial.divByMonic_add_X_sub_C_mul_derivate_divByMonic_eq_derivative {K : Type u_1} [CommRing K] (f : Polynomial K) (a : K) : f /ₘ (X - C a) + (X - C a) * derivative (f /ₘ (X - C a)) = derivative f

Alias of Polynomial.divByMonic_add_X_sub_C_mul_derivative_divByMonic_eq_derivative.

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 : X - C a ∣ f /ₘ (X - C a)) : X - C a ∣ 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' : eval a (derivative f) ≠ 0) : IsCoprime (X - C a) (f /ₘ (X - 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 ∣ p → IsUnit 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.Monic → q.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 Polynomial.leadingCoeff_mul_prod_normalizedFactors {R : Type u} [Field R] [DecidableEq R] (a : Polynomial R) : C a.leadingCoeff * (UniqueFactorizationMonoid.normalizedFactors a).prod = a

The normalized factors of a polynomial over a field times its leading coefficient give the polynomial.

theorem Polynomial.mem_normalizedFactors_iff {R : Type u} [Field R] {p q : Polynomial R} [DecidableEq R] (hq : q ≠ 0) : p ∈ UniqueFactorizationMonoid.normalizedFactors q ↔ Irreducible p ∧ p.Monic ∧ p ∣ q

@[simp]

theorem Polynomial.map_normalize {R : Type u} {S : Type v} [Field R] (p : Polynomial R) [DecidableEq R] [Field S] [DecidableEq S] (f : R →+* S) : map f (normalize p) = normalize (map f p)

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

An irreducible polynomial over a field must have positive degree.