Division of univariate polynomials

The main defs are divByMonic and modByMonic. The compatibility between these is given by modByMonic_add_div. We also define rootMultiplicity.

theorem Polynomial.X_dvd_iff {R : Type u} [Semiring R] {f : Polynomial R} : X ∣ f ↔ f.coeff 0 = 0

theorem Polynomial.X_pow_dvd_iff {R : Type u} [Semiring R] {f : Polynomial R} {n : ℕ} : X ^ n ∣ f ↔ ∀ d < n, f.coeff d = 0

theorem Polynomial.finiteMultiplicity_of_degree_pos_of_monic {R : Type u} [Semiring R] {p q : Polynomial R} (hp : 0 < p.degree) (hmp : p.Monic) (hq : q ≠ 0) : FiniteMultiplicity p q

theorem Polynomial.eq_mul_leadingCoeff_of_monic_of_dvd_of_natDegree_le {R : Type u} [Semiring R] {p q : Polynomial R} (hp : p.Monic) (hdvd : p ∣ q) (hdeg : q.natDegree ≤ p.natDegree) : q = p * C q.leadingCoeff

See Polynomial.eq_leadingCoeff_mul_of_monic_of_dvd_of_natDegree_le for the other multiplication order. That version, unlike this one, requires commutativity.

theorem Polynomial.eq_of_monic_of_dvd_of_natDegree_le {R : Type u} [Semiring R] {p q : Polynomial R} (hp : p.Monic) (hq : q.Monic) (hdvd : p ∣ q) (hdeg : q.natDegree ≤ p.natDegree) : q = p

theorem Polynomial.div_wf_lemma {R : Type u} [Ring R] {p q : Polynomial R} (h : q.degree ≤ p.degree ∧ p ≠ 0) (hq : q.Monic) : (p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree < p.degree

@[irreducible]

noncomputable def Polynomial.divModByMonicAux {R : Type u} [Ring R] (_p : Polynomial R) {q : Polynomial R} : q.Monic → Polynomial R × Polynomial R

See divByMonic.

def Polynomial.divByMonic {R : Type u} [Ring R] (p q : Polynomial R) : Polynomial R

divByMonic, denoted as p /ₘ q, gives the quotient of p by a monic polynomial q.

def Polynomial.modByMonic {R : Type u} [Ring R] (p q : Polynomial R) : Polynomial R

modByMonic, denoted as p %ₘ q, gives the remainder of p by a monic polynomial q.

def Polynomial.«term_/ₘ_» : Lean.TrailingParserDescr

divByMonic, denoted as p /ₘ q, gives the quotient of p by a monic polynomial q.

def Polynomial.«term_%ₘ_» : Lean.TrailingParserDescr

modByMonic, denoted as p %ₘ q, gives the remainder of p by a monic polynomial q.

@[irreducible]

theorem Polynomial.degree_modByMonic_lt {R : Type u} [Ring R] [Nontrivial R] (p : Polynomial R) {q : Polynomial R} (_hq : q.Monic) : (p %ₘ q).degree < q.degree

theorem Polynomial.natDegree_modByMonic_lt {R : Type u} [Ring R] (p : Polynomial R) {q : Polynomial R} (hmq : q.Monic) (hq : q ≠ 1) : (p %ₘ q).natDegree < q.natDegree

@[simp]

theorem Polynomial.zero_modByMonic {R : Type u} [Ring R] (p : Polynomial R) : 0 %ₘ p = 0

@[simp]

theorem Polynomial.zero_divByMonic {R : Type u} [Ring R] (p : Polynomial R) : 0 /ₘ p = 0

@[simp]

theorem Polynomial.modByMonic_zero {R : Type u} [Ring R] (p : Polynomial R) : p %ₘ 0 = p

@[simp]

theorem Polynomial.divByMonic_zero {R : Type u} [Ring R] (p : Polynomial R) : p /ₘ 0 = 0

theorem Polynomial.divByMonic_eq_of_not_monic {R : Type u} [Ring R] {q : Polynomial R} (p : Polynomial R) (hq : ¬q.Monic) : p /ₘ q = 0

theorem Polynomial.modByMonic_eq_of_not_monic {R : Type u} [Ring R] {q : Polynomial R} (p : Polynomial R) (hq : ¬q.Monic) : p %ₘ q = p

theorem Polynomial.modByMonic_eq_self_iff {R : Type u} [Ring R] {p q : Polynomial R} [Nontrivial R] (hq : q.Monic) : p %ₘ q = p ↔ p.degree < q.degree

theorem Polynomial.degree_modByMonic_le {R : Type u} [Ring R] (p : Polynomial R) {q : Polynomial R} (hq : q.Monic) : (p %ₘ q).degree ≤ q.degree

theorem Polynomial.degree_modByMonic_le_left {R : Type u} [Ring R] {p q : Polynomial R} : (p %ₘ q).degree ≤ p.degree

theorem Polynomial.natDegree_modByMonic_le {R : Type u} [Ring R] (p : Polynomial R) {g : Polynomial R} (hg : g.Monic) : (p %ₘ g).natDegree ≤ g.natDegree

theorem Polynomial.natDegree_modByMonic_le_left {R : Type u} [Ring R] {p q : Polynomial R} : (p %ₘ q).natDegree ≤ p.natDegree

theorem Polynomial.X_dvd_sub_C {R : Type u} [Ring R] {p : Polynomial R} : X ∣ p - C (p.coeff 0)

@[irreducible]

theorem Polynomial.modByMonic_eq_sub_mul_div {R : Type u} [Ring R] (p : Polynomial R) {q : Polynomial R} (_hq : q.Monic) : p %ₘ q = p - q * (p /ₘ q)

theorem Polynomial.modByMonic_add_div {R : Type u} [Ring R] (p : Polynomial R) {q : Polynomial R} (hq : q.Monic) : p %ₘ q + q * (p /ₘ q) = p

theorem Polynomial.divByMonic_eq_zero_iff {R : Type u} [Ring R] {p q : Polynomial R} [Nontrivial R] (hq : q.Monic) : p /ₘ q = 0 ↔ p.degree < q.degree

theorem Polynomial.degree_add_divByMonic {R : Type u} [Ring R] {p q : Polynomial R} (hq : q.Monic) (h : q.degree ≤ p.degree) : q.degree + (p /ₘ q).degree = p.degree

theorem Polynomial.degree_divByMonic_le {R : Type u} [Ring R] (p q : Polynomial R) : (p /ₘ q).degree ≤ p.degree

theorem Polynomial.degree_divByMonic_lt {R : Type u} [Ring R] (p : Polynomial R) {q : Polynomial R} (hq : q.Monic) (hp0 : p ≠ 0) (h0q : 0 < q.degree) : (p /ₘ q).degree < p.degree

theorem Polynomial.natDegree_divByMonic {R : Type u} [Ring R] (f : Polynomial R) {g : Polynomial R} (hg : g.Monic) : (f /ₘ g).natDegree = f.natDegree - g.natDegree

theorem Polynomial.div_modByMonic_unique {R : Type u} [Ring R] {f g : Polynomial R} (q r : Polynomial R) (hg : g.Monic) (h : r + g * q = f ∧ r.degree < g.degree) : f /ₘ g = q ∧ f %ₘ g = r

theorem Polynomial.map_mod_divByMonic {R : Type u} {S : Type v} [Ring R] {p q : Polynomial R} [Ring S] (f : R →+* S) (hq : q.Monic) : map f (p /ₘ q) = map f p /ₘ map f q ∧ map f (p %ₘ q) = map f p %ₘ map f q

theorem Polynomial.map_divByMonic {R : Type u} {S : Type v} [Ring R] {p q : Polynomial R} [Ring S] (f : R →+* S) (hq : q.Monic) : map f (p /ₘ q) = map f p /ₘ map f q

theorem Polynomial.map_modByMonic {R : Type u} {S : Type v} [Ring R] {p q : Polynomial R} [Ring S] (f : R →+* S) (hq : q.Monic) : map f (p %ₘ q) = map f p %ₘ map f q

theorem Polynomial.modByMonic_eq_zero_iff_dvd {R : Type u} [Ring R] {p q : Polynomial R} (hq : q.Monic) : p %ₘ q = 0 ↔ q ∣ p

@[simp]

theorem Polynomial.self_mul_modByMonic {R : Type u} [Ring R] {p q : Polynomial R} (hq : q.Monic) : q * p %ₘ q = 0

See Polynomial.mul_self_modByMonic for the other multiplication order. That version, unlike this one, requires commutativity.

theorem Polynomial.map_dvd_map {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) (hf : Function.Injective ⇑f) {x y : Polynomial R} (hx : x.Monic) : map f x ∣ map f y ↔ x ∣ y

@[simp]

theorem Polynomial.modByMonic_one {R : Type u} [Ring R] (p : Polynomial R) : p %ₘ 1 = 0

@[simp]

theorem Polynomial.divByMonic_one {R : Type u} [Ring R] (p : Polynomial R) : p /ₘ 1 = p

theorem Polynomial.sum_modByMonic_coeff {R : Type u} [Ring R] {p q : Polynomial R} (hq : q.Monic) {n : ℕ} (hn : q.degree ≤ ↑n) : ∑ i : Fin n, (monomial ↑i) ((p %ₘ q).coeff ↑i) = p %ₘ q

theorem Polynomial.mul_divByMonic_cancel_left {R : Type u} [Ring R] (p : Polynomial R) {q : Polynomial R} (hmo : q.Monic) : q * p /ₘ q = p

theorem Polynomial.coeff_divByMonic_X_sub_C_rec {R : Type u} [Ring R] (p : Polynomial R) (a : R) (n : ℕ) : (p /ₘ (X - C a)).coeff n = p.coeff (n + 1) + a * (p /ₘ (X - C a)).coeff (n + 1)

theorem Polynomial.coeff_divByMonic_X_sub_C {R : Type u} [Ring R] (p : Polynomial R) (a : R) (n : ℕ) : (p /ₘ (X - C a)).coeff n = ∑ i ∈ Finset.Icc (n + 1) p.natDegree, a ^ (i - (n + 1)) * p.coeff i

theorem Polynomial.not_isField (R : Type u) [Ring R] : ¬IsField (Polynomial R)

@[deprecated Classical.dec (since := "2026-02-07")]

def Polynomial.decidableDvdMonic {R : Type u} [Ring R] {q : Polynomial R} [DecidableEq R] (p : Polynomial R) (hq : q.Monic) : Decidable (q ∣ p)

An algorithm for deciding polynomial divisibility. Prefer Classical.dec, as the algorithm relies on %ₘ and so is noncomputable.

theorem Polynomial.finiteMultiplicity_X_sub_C {R : Type u} [Ring R] {p : Polynomial R} (a : R) (h0 : p ≠ 0) : FiniteMultiplicity (X - C a) p

def Polynomial.rootMultiplicity {R : Type u} [Ring R] (a : R) (p : Polynomial R) : ℕ

The largest power of X - C a which divides p.

theorem Polynomial.rootMultiplicity_eq_natFind_of_ne_zero {R : Type u} [Ring R] {p : Polynomial R} (p0 : p ≠ 0) {a : R} [DecidablePred fun (n : ℕ) => ¬(X - C a) ^ (n + 1) ∣ p] : rootMultiplicity a p = Nat.find ⋯

@[deprecated Polynomial.rootMultiplicity_eq_natFind_of_ne_zero (since := "2026-02-12")]

theorem Polynomial.rootMultiplicity_eq_nat_find_of_nonzero {R : Type u} [Ring R] {p : Polynomial R} (p0 : p ≠ 0) {a : R} [DecidablePred fun (n : ℕ) => ¬(X - C a) ^ (n + 1) ∣ p] : rootMultiplicity a p = Nat.find ⋯

Alias of Polynomial.rootMultiplicity_eq_natFind_of_ne_zero.

theorem Polynomial.rootMultiplicity_eq_multiplicity {R : Type u} [Ring R] [DecidableEq R] (p : Polynomial R) (a : R) : rootMultiplicity a p = if p = 0 then 0 else multiplicity (X - C a) p

@[simp]

theorem Polynomial.rootMultiplicity_zero {R : Type u} [Ring R] {x : R} : rootMultiplicity x 0 = 0

@[simp]

theorem Polynomial.rootMultiplicity_C {R : Type u} [Ring R] (r a : R) : rootMultiplicity a (C r) = 0

theorem Polynomial.pow_rootMultiplicity_dvd {R : Type u} [Ring R] (p : Polynomial R) (a : R) : (X - C a) ^ rootMultiplicity a p ∣ p

theorem Polynomial.pow_mul_divByMonic_rootMultiplicity_eq {R : Type u} [Ring R] (p : Polynomial R) (a : R) : (X - C a) ^ rootMultiplicity a p * (p /ₘ (X - C a) ^ rootMultiplicity a p) = p

theorem Polynomial.exists_eq_pow_rootMultiplicity_mul_and_not_dvd {R : Type u} [Ring R] (p : Polynomial R) (hp : p ≠ 0) (a : R) : ∃ (q : Polynomial R), p = (X - C a) ^ rootMultiplicity a p * q ∧ ¬X - C a ∣ q

@[simp]

theorem Polynomial.modByMonic_X_sub_C_eq_C_eval {R : Type u} [CommRing R] (p : Polynomial R) (a : R) : p %ₘ (X - C a) = C (eval a p)

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

theorem Polynomial.dvd_iff_isRoot {R : Type u} {a : R} [CommRing R] {p : Polynomial R} : X - C a ∣ p ↔ p.IsRoot a

theorem Polynomial.X_sub_C_dvd_sub_C_eval {R : Type u} {a : R} [CommRing R] {p : Polynomial R} : X - C a ∣ p - C (eval a p)

theorem Polynomial.modByMonic_X {R : Type u} [CommRing R] (p : Polynomial R) : p %ₘ X = C (eval 0 p)

theorem Polynomial.eval₂_modByMonic_eq_self_of_root {R : Type u} {S : Type v} [CommRing R] [CommRing S] {f : R →+* S} {p q : Polynomial R} (hq : q.Monic) {x : S} (hx : eval₂ f x q = 0) : eval₂ f x (p %ₘ q) = eval₂ f x p

theorem Polynomial.sub_dvd_eval_sub {R : Type u} [CommRing R] (a b : R) (p : Polynomial R) : a - b ∣ eval a p - eval b p

theorem Polynomial.IsRoot.dvd_coeff_zero {R : Type u} [CommRing R] {p : Polynomial R} {x : R} (h : p.IsRoot x) : x ∣ p.coeff 0

@[simp]

theorem Polynomial.rootMultiplicity_eq_zero_iff {R : Type u} [CommRing R] {p : Polynomial R} {x : R} : rootMultiplicity x p = 0 ↔ p.IsRoot x → p = 0

theorem Polynomial.rootMultiplicity_eq_zero {R : Type u} [CommRing R] {p : Polynomial R} {x : R} (h : ¬p.IsRoot x) : rootMultiplicity x p = 0

@[simp]

theorem Polynomial.rootMultiplicity_pos' {R : Type u} [CommRing R] {p : Polynomial R} {x : R} : 0 < rootMultiplicity x p ↔ p ≠ 0 ∧ p.IsRoot x

theorem Polynomial.rootMultiplicity_pos {R : Type u} [CommRing R] {p : Polynomial R} (hp : p ≠ 0) {x : R} : 0 < rootMultiplicity x p ↔ p.IsRoot x

theorem Polynomial.eval_divByMonic_pow_rootMultiplicity_ne_zero {R : Type u} [CommRing R] {p : Polynomial R} (a : R) (hp : p ≠ 0) : eval a (p /ₘ (X - C a) ^ rootMultiplicity a p) ≠ 0

@[simp]

theorem Polynomial.mul_self_modByMonic {R : Type u} [CommRing R] {p q : Polynomial R} (hq : q.Monic) : p * q %ₘ q = 0

See Polynomial.self_mul_modByMonic for the other multiplication order. This version, unlike that one, requires commutativity.

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

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

theorem Polynomial.neg_modByMonic {R : Type u} [CommRing R] (p q : Polynomial R) : -p %ₘ q = -(p %ₘ q)

theorem Polynomial.sub_modByMonic {R : Type u} [CommRing R] (p₁ p₂ q : Polynomial R) : (p₁ - p₂) %ₘ q = p₁ %ₘ q - p₂ %ₘ q

theorem Polynomial.mul_modByMonic {R : Type u} [CommRing R] (p₁ p₂ q : Polynomial R) : p₁ * p₂ %ₘ q = p₁ %ₘ q * (p₂ %ₘ q) %ₘ q

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

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

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

theorem Polynomial.rootMultiplicity_le_rootMultiplicity_of_dvd {R : Type u} [CommRing R] {p q : Polynomial R} (hq : q ≠ 0) (hpq : p ∣ q) (x : R) : rootMultiplicity x p ≤ rootMultiplicity x q

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

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

See Polynomial.rootMultiplicity_eq_natTrailingDegree for the general case.

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

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

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

theorem Irreducible.not_isRoot_of_natDegree_ne_one {R : Type u} [CommRing R] {p : Polynomial R} [IsDomain R] (hi : Irreducible p) (hdeg : p.natDegree ≠ 1) {x : R} : ¬p.IsRoot x

theorem Irreducible.isRoot_eq_bot_of_natDegree_ne_one {R : Type u} [CommRing R] {p : Polynomial R} [IsDomain R] (hi : Irreducible p) (hdeg : p.natDegree ≠ 1) : p.IsRoot = ⊥

theorem Irreducible.subsingleton_isRoot {R : Type u} [CommRing R] {p : Polynomial R} [IsDomain R] [IsLeftCancelMulZero R] (hi : Irreducible p) : {x : R | p.IsRoot x}.Subsingleton

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

theorem Polynomial.eq_of_dvd_of_natDegree_le_of_leadingCoeff {R : Type u} [CommRing R] [IsDomain R] {p 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 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 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 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} [CommSemiring R] {p q : Polynomial R} (hp : p.Monic) (hdvd : p ∣ q) (hdeg : q.natDegree ≤ p.natDegree) : q = C q.leadingCoeff * p