# mathlibdocumentation

data.polynomial.ring_division

# Theory of univariate polynomials #

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

theorem polynomial.nat_degree_pos_of_aeval_root {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] [ S] {p : polynomial R} (hp : p 0) {z : S} (hz : p = 0) (inj : ∀ (x : R), S) x = 0x = 0) :
theorem polynomial.degree_pos_of_aeval_root {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] [ S] {p : polynomial R} (hp : p 0) {z : S} (hz : p = 0) (inj : ∀ (x : R), S) x = 0x = 0) :
0 < p.degree
theorem polynomial.aeval_mod_by_monic_eq_self_of_root {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] [ S] {p q : polynomial R} (hq : q.monic) {x : S} (hx : q = 0) :
(p %ₘ q) = p
@[instance]
def polynomial.no_zero_divisors {R : Type u} [comm_ring R]  :
theorem polynomial.nat_degree_mul {R : Type u} [comm_ring R] {p q : polynomial R} (hp : p 0) (hq : q 0) :
(p * q).nat_degree =
@[simp]
theorem polynomial.nat_degree_pow {R : Type u} [comm_ring R] (p : polynomial R) (n : ) :
theorem polynomial.root_mul {R : Type u} {a : R} [comm_ring R] {p q : polynomial R} :
(p * q).is_root a p.is_root a q.is_root a
theorem polynomial.root_or_root_of_root_mul {R : Type u} {a : R} [comm_ring R] {p q : polynomial R} (h : (p * q).is_root a) :
theorem polynomial.degree_le_mul_left {R : Type u} [comm_ring R] {q : polynomial R} (p : polynomial R) (hq : q 0) :
p.degree (p * q).degree
theorem polynomial.nat_degree_le_of_dvd {R : Type u} [comm_ring R] {p q : polynomial R} (h1 : p q) (h2 : q 0) :
@[instance]
def polynomial.integral_domain {R : Type u}  :
Equations
theorem polynomial.nat_trailing_degree_mul {R : Type u} {p q : polynomial R} (hp : p 0) (hq : q 0) :
theorem polynomial.degree_eq_zero_of_is_unit {R : Type u} {p : polynomial R} (h : is_unit p) :
p.degree = 0
@[simp]
theorem polynomial.degree_coe_units {R : Type u} (u : units (polynomial R)) :
theorem polynomial.prime_X_sub_C {R : Type u} {r : R} :
theorem polynomial.prime_X {R : Type u}  :
theorem polynomial.prime_of_degree_eq_one_of_monic {R : Type u} {p : polynomial R} (hp1 : p.degree = 1) (hm : p.monic) :
theorem polynomial.irreducible_X_sub_C {R : Type u} (r : R) :
theorem polynomial.irreducible_X {R : Type u}  :
theorem polynomial.irreducible_of_degree_eq_one_of_monic {R : Type u} {p : polynomial R} (hp1 : p.degree = 1) (hm : p.monic) :
theorem polynomial.eq_of_monic_of_associated {R : Type u} {p q : polynomial R} (hp : p.monic) (hq : q.monic) (hpq : q) :
p = q
@[simp]
theorem polynomial.root_multiplicity_zero {R : Type u} {x : R} :
theorem polynomial.root_multiplicity_eq_zero {R : Type u} {p : polynomial R} {x : R} (h : ¬p.is_root x) :
theorem polynomial.root_multiplicity_pos {R : Type u} {p : polynomial R} (hp : p 0) {x : R} :
theorem polynomial.root_multiplicity_mul {R : Type u} {p q : polynomial R} {x : R} (hpq : p * q 0) :
theorem polynomial.root_multiplicity_X_sub_C_self {R : Type u} {x : R} :
theorem polynomial.root_multiplicity_X_sub_C {R : Type u} {x y : R} :
= ite (x = y) 1 0
theorem polynomial.root_multiplicity_X_sub_C_pow {R : Type u} (a : R) (n : ) :
= n

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

theorem polynomial.root_multiplicity_of_dvd {R : Type u} {p : polynomial R} {a : R} {n : } (hzero : p 0) (h : ^ n p) :

If (X - a) ^ n divides a polynomial p then the multiplicity of a as root of p is at least n.

theorem polynomial.root_multiplicity_add {R : Type u} {p q : polynomial R} (a : R) (hzero : p + q 0) :
(p + q)

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

theorem polynomial.exists_multiset_roots {R : Type u} {p : polynomial R} (hp : p 0) :
∃ (s : multiset R), p.degree ∀ (a : R),
def polynomial.roots {R : Type u} (p : polynomial R) :

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

Equations
@[simp]
theorem polynomial.roots_zero {R : Type u}  :
0.roots = 0
theorem polynomial.card_roots {R : Type u} {p : polynomial R} (hp0 : p 0) :
theorem polynomial.card_roots' {R : Type u} {p : polynomial R} (hp0 : p 0) :
theorem polynomial.card_roots_sub_C {R : Type u} {p : polynomial R} {a : R} (hp0 : 0 < p.degree) :
theorem polynomial.card_roots_sub_C' {R : Type u} {p : polynomial R} {a : R} (hp0 : 0 < p.degree) :
@[simp]
theorem polynomial.count_roots {R : Type u} {a : R} {p : polynomial R} (hp : p 0) :
@[simp]
theorem polynomial.mem_roots {R : Type u} {a : R} {p : polynomial R} (hp : p 0) :
theorem polynomial.eq_zero_of_infinite_is_root {R : Type u} (p : polynomial R) (h : {x : R | p.is_root x}.infinite) :
p = 0
theorem polynomial.exists_max_root {R : Type u} [linear_order R] (p : polynomial R) (hp : p 0) :
∃ (x₀ : R), ∀ (x : R), p.is_root xx x₀
theorem polynomial.exists_min_root {R : Type u} [linear_order R] (p : polynomial R) (hp : p 0) :
∃ (x₀ : R), ∀ (x : R), p.is_root xx₀ x
theorem polynomial.eq_of_infinite_eval_eq {R : Type u_1} (p q : polynomial R) (h : {x : R | = q}.infinite) :
p = q
theorem polynomial.roots_mul {R : Type u} {p q : polynomial R} (hpq : p * q 0) :
(p * q).roots = p.roots + q.roots
@[simp]
theorem polynomial.mem_roots_sub_C {R : Type u} {p : polynomial R} {a x : R} (hp0 : 0 < p.degree) :
x (p - .roots = a
@[simp]
theorem polynomial.roots_X_sub_C {R : Type u} (r : R) :
= r ::ₘ 0
@[simp]
theorem polynomial.roots_C {R : Type u} (x : R) :
.roots = 0
@[simp]
theorem polynomial.roots_one {R : Type u}  :
theorem polynomial.roots_list_prod {R : Type u} (L : list (polynomial R)) :
0 L
theorem polynomial.roots_multiset_prod {R : Type u} (m : multiset (polynomial R)) :
0 m
theorem polynomial.roots_prod {R : Type u} {ι : Type u_1} (f : ι → ) (s : finset ι) :
s.prod f 0(s.prod f).roots = s.val.bind (λ (i : ι), (f i).roots)
theorem polynomial.roots_prod_X_sub_C {R : Type u} (s : finset R) :
(∏ (a : R) in s, .roots = s.val
theorem polynomial.card_roots_X_pow_sub_C {R : Type u} {n : } (hn : 0 < n) (a : R) :
n
def polynomial.nth_roots {R : Type u} (n : ) (a : R) :

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

Equations
@[simp]
theorem polynomial.mem_nth_roots {R : Type u} {n : } (hn : 0 < n) {a x : R} :
x ^ n = a
@[simp]
theorem polynomial.nth_roots_zero {R : Type u} (r : R) :
theorem polynomial.card_nth_roots {R : Type u} (n : ) (a : R) :
def polynomial.nth_roots_finset (n : ) (R : Type u_1)  :

The multiset nth_roots ↑n (1 : R) as a finset.

Equations
@[simp]
theorem polynomial.mem_nth_roots_finset {R : Type u} {n : } (h : 0 < n) {x : R} :
x ^ n = 1
theorem polynomial.coeff_comp_degree_mul_degree {R : Type u} {p q : polynomial R} (hqd0 : q.nat_degree 0) :
theorem polynomial.nat_degree_comp {R : Type u} {p q : polynomial R} :
theorem polynomial.leading_coeff_comp {R : Type u} {p q : polynomial R} (hq : q.nat_degree 0) :
theorem polynomial.units_coeff_zero_smul {R : Type u} (c : units (polynomial R)) (p : polynomial R) :
c.coeff 0 p = (c) * p
@[simp]
theorem polynomial.nat_degree_coe_units {R : Type u} (u : units (polynomial R)) :
theorem polynomial.comp_eq_zero_iff {R : Type u} {p 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} [infinite R] (p : polynomial R) (h : ∀ (x : R), = 0) :
p = 0
theorem polynomial.funext {R : Type u} [infinite R] {p q : polynomial R} (ext : ∀ (r : R), = ) :
p = q
def polynomial.root_set {R : Type u} (p : polynomial R) (S : Type u_1) [ S] :
set S

The set of distinct roots of p in E.

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

Equations
theorem polynomial.root_set_def {R : Type u} (p : polynomial R) (S : Type u_1) [ S] :
@[simp]
theorem polynomial.root_set_zero {R : Type u} (S : Type u_1) [ S] :
@[simp]
theorem polynomial.root_set_C {R : Type u} {S : Type v} [ S] (a : R) :
S =
@[instance]
def polynomial.root_set_fintype {R : Type u_1} (p : polynomial R) (S : Type u_2) [ S] :
Equations
theorem polynomial.root_set_finite {R : Type u_1} (p : polynomial R) (S : Type u_2) [ S] :
theorem polynomial.is_unit_iff {R : Type u} {f : polynomial R} :
∃ (r : R),
theorem polynomial.degree_eq_degree_of_associated {R : Type u} {p q : polynomial R} (h : q) :
theorem polynomial.degree_eq_one_of_irreducible_of_root {R : Type u} {p : polynomial R} (hi : irreducible p) {x : R} (hx : p.is_root x) :
p.degree = 1
theorem polynomial.leading_coeff_div_by_monic_of_monic {R : Type u} {p q : polynomial R} (hmonic : q.monic) (hdegree : q.degree p.degree) :

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

theorem polynomial.eq_of_monic_of_dvd_of_nat_degree_le {R : Type u} {p q : polynomial R} (hp : p.monic) (hq : q.monic) (hdiv : p q) (hdeg : q.nat_degree p.nat_degree) :
q = p
theorem polynomial.is_unit_of_is_unit_leading_coeff_of_is_unit_map {R : Type u} {S : Type v} [semiring R] (φ : R →+* S) (f : polynomial R) (hf : is_unit f.leading_coeff) (H : is_unit f)) :
theorem polynomial.monic.irreducible_of_irreducible_map {R : Type u} {S : Type v} (φ : R →+* S) (f : polynomial R) (h_mon : f.monic) (h_irr : 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.

theorem is_integral_domain.polynomial {R : Type u_1} [comm_ring R] (h : is_integral_domain R) :

Lift evidence that is_integral_domain R to is_integral_domain (polynomial R).