data.polynomial.ring_division

source

theorem polynomial.nat_degree_pos_of_aeval_root {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] [algebra R S] {p : polynomial R} (hp : p ≠ 0) {z : S} (hz : ⇑(polynomial.aeval z) p = 0) (inj : ∀ (x : R), ⇑(algebra_map R S) x = 0 → x = 0) :

0 < p.nat_degree

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theorem polynomial.degree_pos_of_aeval_root {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] [algebra R S] {p : polynomial R} (hp : p ≠ 0) {z : S} (hz : ⇑(polynomial.aeval z) p = 0) (inj : ∀ (x : R), ⇑(algebra_map R S) x = 0 → x = 0) :

0 < p.degree

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theorem polynomial.aeval_mod_by_monic_eq_self_of_root {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] [algebra R S] {p q : polynomial R} (hq : q.monic) {x : S} (hx : ⇑(polynomial.aeval x) q = 0) :

⇑(polynomial.aeval x) (p %ₘ q) = ⇑(polynomial.aeval x) p

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theorem polynomial.mod_by_monic_eq_of_dvd_sub {R : Type u} [comm_ring R] {q : polynomial R} [nontrivial R] (hq : q.monic) {p₁ p₂ : polynomial R} (h : q ∣ p₁ - p₂) :

p₁ %ₘ q = p₂ %ₘ q

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theorem polynomial.add_mod_by_monic {R : Type u} [comm_ring R] {q : polynomial R} [nontrivial R] (hq : q.monic) (p₁ p₂ : polynomial R) :

(p₁ + p₂) %ₘ q = p₁ %ₘ q + p₂ %ₘ q

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theorem polynomial.smul_mod_by_monic {R : Type u} [comm_ring R] {q : polynomial R} [nontrivial R] (hq : q.monic) (c : R) (p : polynomial R) :

c • p %ₘ q = c • (p %ₘ q)

source

noncomputable def polynomial.mod_by_monic_hom {R : Type u} [comm_ring R] {q : polynomial R} [nontrivial R] (hq : q.monic) :

polynomial R →ₗ[R] polynomial R

polynomial.mod_by_monic_hom (hq : monic (q : polynomial R)) is _ %ₘ q as a R-linear map.

Equations

polynomial.mod_by_monic_hom hq = {to_fun := λ (p : polynomial R), p %ₘ q, map_add' := _, map_smul' := _}

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@[simp]

theorem polynomial.mod_by_monic_hom_apply {R : Type u} [comm_ring R] {q : polynomial R} [nontrivial R] (hq : q.monic) (p : polynomial R) :

⇑(polynomial.mod_by_monic_hom hq) p = p %ₘ q

source

@[protected, instance]

def polynomial.no_zero_divisors {R : Type u} [ring R] [no_zero_divisors R] :

no_zero_divisors (polynomial R)

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theorem polynomial.nat_degree_mul {R : Type u} [ring R] [no_zero_divisors R] {p q : polynomial R} (hp : p ≠ 0) (hq : q ≠ 0) :

(p * q).nat_degree = p.nat_degree + q.nat_degree

source

@[simp]

theorem polynomial.nat_degree_pow {R : Type u} [ring R] [no_zero_divisors R] (p : polynomial R) (n : ℕ) :

(p ^ n).nat_degree = n * p.nat_degree

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theorem polynomial.degree_le_mul_left {R : Type u} [ring R] [no_zero_divisors R] {q : polynomial R} (p : polynomial R) (hq : q ≠ 0) :

p.degree ≤ (p * q).degree

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theorem polynomial.nat_degree_le_of_dvd {R : Type u} [ring R] [no_zero_divisors R] {p q : polynomial R} (h1 : p ∣ q) (h2 : q ≠ 0) :

p.nat_degree ≤ q.nat_degree

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theorem polynomial.root_mul {R : Type u} {a : R} [comm_ring R] [no_zero_divisors R] {p q : polynomial R} :

(p * q).is_root a ↔ p.is_root a ∨ q.is_root a

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theorem polynomial.root_or_root_of_root_mul {R : Type u} {a : R} [comm_ring R] [no_zero_divisors R] {p q : polynomial R} (h : (p * q).is_root a) :

p.is_root a ∨ q.is_root a

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@[protected, instance]

def polynomial.is_domain {R : Type u} [ring R] [is_domain R] :

is_domain (polynomial R)

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theorem polynomial.nat_trailing_degree_mul {R : Type u} [ring R] [is_domain R] {p q : polynomial R} (hp : p ≠ 0) (hq : q ≠ 0) :

(p * q).nat_trailing_degree = p.nat_trailing_degree + q.nat_trailing_degree

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theorem polynomial.degree_eq_zero_of_is_unit {R : Type u} [comm_ring R] [is_domain R] {p : polynomial R} (h : is_unit p) :

p.degree = 0

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@[simp]

theorem polynomial.degree_coe_units {R : Type u} [comm_ring R] [is_domain R] (u : (polynomial R)ˣ) :

↑u.degree = 0

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theorem polynomial.prime_X_sub_C {R : Type u} [comm_ring R] [is_domain R] (r : R) :

prime (polynomial.X - ⇑polynomial.C r)

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theorem polynomial.prime_X {R : Type u} [comm_ring R] [is_domain R] :

prime polynomial.X

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theorem polynomial.monic.prime_of_degree_eq_one {R : Type u} [comm_ring R] [is_domain R] {p : polynomial R} (hp1 : p.degree = 1) (hm : p.monic) :

prime p

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theorem polynomial.irreducible_X_sub_C {R : Type u} [comm_ring R] [is_domain R] (r : R) :

irreducible (polynomial.X - ⇑polynomial.C r)

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theorem polynomial.irreducible_X {R : Type u} [comm_ring R] [is_domain R] :

irreducible polynomial.X

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theorem polynomial.monic.irreducible_of_degree_eq_one {R : Type u} [comm_ring R] [is_domain R] {p : polynomial R} (hp1 : p.degree = 1) (hm : p.monic) :

irreducible p

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theorem polynomial.eq_of_monic_of_associated {R : Type u} [comm_ring R] [is_domain R] {p q : polynomial R} (hp : p.monic) (hq : q.monic) (hpq : associated p q) :

p = q

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theorem polynomial.root_multiplicity_mul {R : Type u} [comm_ring R] [is_domain R] {p q : polynomial R} {x : R} (hpq : p * q ≠ 0) :

polynomial.root_multiplicity x (p * q) = polynomial.root_multiplicity x p + polynomial.root_multiplicity x q

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theorem polynomial.root_multiplicity_X_sub_C_self {R : Type u} [comm_ring R] [is_domain R] {x : R} :

polynomial.root_multiplicity x (polynomial.X - ⇑polynomial.C x) = 1

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theorem polynomial.root_multiplicity_X_sub_C {R : Type u} [comm_ring R] [is_domain R] {x y : R} :

polynomial.root_multiplicity x (polynomial.X - ⇑polynomial.C y) = ite (x = y) 1 0

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theorem polynomial.root_multiplicity_X_sub_C_pow {R : Type u} [comm_ring R] [is_domain R] (a : R) (n : ℕ) :

polynomial.root_multiplicity a ((polynomial.X - ⇑polynomial.C a) ^ n) = n

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

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theorem polynomial.root_multiplicity_of_dvd {R : Type u} [comm_ring R] [is_domain R] {p : polynomial R} {a : R} {n : ℕ} (hzero : p ≠ 0) (h : (polynomial.X - ⇑polynomial.C a) ^ n ∣ p) :

n ≤ polynomial.root_multiplicity a p

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

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theorem polynomial.root_multiplicity_add {R : Type u} [comm_ring R] [is_domain R] {p q : polynomial R} (a : R) (hzero : p + q ≠ 0) :

min (polynomial.root_multiplicity a p) (polynomial.root_multiplicity a q) ≤ polynomial.root_multiplicity a (p + q)

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

source

theorem polynomial.exists_multiset_roots {R : Type u} [comm_ring R] [is_domain R] {p : polynomial R} (hp : p ≠ 0) :

∃ (s : multiset R), ↑(⇑multiset.card s) ≤ p.degree ∧ ∀ (a : R), multiset.count a s = polynomial.root_multiplicity a p

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noncomputable def polynomial.roots {R : Type u} [comm_ring R] [is_domain R] (p : polynomial R) :

multiset R

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

Equations

p.roots = dite (p = 0) (λ (h : p = 0), ∅) (λ (h : ¬p = 0), classical.some _)

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@[simp]

theorem polynomial.roots_zero {R : Type u} [comm_ring R] [is_domain R] :

0.roots = 0

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theorem polynomial.card_roots {R : Type u} [comm_ring R] [is_domain R] {p : polynomial R} (hp0 : p ≠ 0) :

↑(⇑multiset.card p.roots) ≤ p.degree

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theorem polynomial.card_roots' {R : Type u} [comm_ring R] [is_domain R] (p : polynomial R) :

⇑multiset.card p.roots ≤ p.nat_degree

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theorem polynomial.card_roots_sub_C {R : Type u} [comm_ring R] [is_domain R] {p : polynomial R} {a : R} (hp0 : 0 < p.degree) :

↑(⇑multiset.card (p - ⇑polynomial.C a).roots) ≤ p.degree

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theorem polynomial.card_roots_sub_C' {R : Type u} [comm_ring R] [is_domain R] {p : polynomial R} {a : R} (hp0 : 0 < p.degree) :

⇑multiset.card (p - ⇑polynomial.C a).roots ≤ p.nat_degree

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@[simp]

theorem polynomial.count_roots {R : Type u} {a : R} [comm_ring R] [is_domain R] (p : polynomial R) :

multiset.count a p.roots = polynomial.root_multiplicity a p

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@[simp]

theorem polynomial.mem_roots {R : Type u} {a : R} [comm_ring R] [is_domain R] {p : polynomial R} (hp : p ≠ 0) :

a ∈ p.roots ↔ p.is_root a

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theorem polynomial.card_le_degree_of_subset_roots {R : Type u} [comm_ring R] [is_domain R] {p : polynomial R} {Z : finset R} (h : Z.val ⊆ p.roots) :

Z.card ≤ p.nat_degree

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theorem polynomial.eq_zero_of_infinite_is_root {R : Type u} [comm_ring R] [is_domain R] (p : polynomial R) (h : {x : R | p.is_root x}.infinite) :

p = 0

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theorem polynomial.exists_max_root {R : Type u} [comm_ring R] [is_domain R] [linear_order R] (p : polynomial R) (hp : p ≠ 0) :

∃ (x₀ : R), ∀ (x : R), p.is_root x → x ≤ x₀

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theorem polynomial.exists_min_root {R : Type u} [comm_ring R] [is_domain R] [linear_order R] (p : polynomial R) (hp : p ≠ 0) :

∃ (x₀ : R), ∀ (x : R), p.is_root x → x₀ ≤ x

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theorem polynomial.eq_of_infinite_eval_eq {R : Type u_1} [comm_ring R] [is_domain R] (p q : polynomial R) (h : {x : R | polynomial.eval x p = polynomial.eval x q}.infinite) :

p = q

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theorem polynomial.roots_mul {R : Type u} [comm_ring R] [is_domain R] {p q : polynomial R} (hpq : p * q ≠ 0) :

(p * q).roots = p.roots + q.roots

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theorem polynomial.roots.le_of_dvd {R : Type u} [comm_ring R] [is_domain R] {p q : polynomial R} (h : q ≠ 0) :

p ∣ q → p.roots ≤ q.roots

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@[simp]

theorem polynomial.mem_roots_sub_C {R : Type u} [comm_ring R] [is_domain R] {p : polynomial R} {a x : R} (hp0 : 0 < p.degree) :

x ∈ (p - ⇑polynomial.C a).roots ↔ polynomial.eval x p = a

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@[simp]

theorem polynomial.roots_X_sub_C {R : Type u} [comm_ring R] [is_domain R] (r : R) :

(polynomial.X - ⇑polynomial.C r).roots = {r}

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@[simp]

theorem polynomial.roots_C {R : Type u} [comm_ring R] [is_domain R] (x : R) :

(⇑polynomial.C x).roots = 0

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@[simp]

theorem polynomial.roots_one {R : Type u} [comm_ring R] [is_domain R] :

1.roots = ∅

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theorem polynomial.roots_smul_nonzero {R : Type u} [comm_ring R] [is_domain R] (p : polynomial R) {r : R} (hr : r ≠ 0) :

(r • p).roots = p.roots

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theorem polynomial.roots_list_prod {R : Type u} [comm_ring R] [is_domain R] (L : list (polynomial R)) :

0 ∉ L → L.prod.roots = ↑L.bind polynomial.roots

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theorem polynomial.roots_multiset_prod {R : Type u} [comm_ring R] [is_domain R] (m : multiset (polynomial R)) :

0 ∉ m → m.prod.roots = m.bind polynomial.roots

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theorem polynomial.roots_prod {R : Type u} [comm_ring R] [is_domain R] {ι : Type u_1} (f : ι → polynomial R) (s : finset ι) :

s.prod f ≠ 0 → (s.prod f).roots = s.val.bind (λ (i : ι), (f i).roots)

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theorem polynomial.roots_prod_X_sub_C {R : Type u} [comm_ring R] [is_domain R] (s : finset R) :

(∏ (a : R) in s, (polynomial.X - ⇑polynomial.C a)).roots = s.val

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theorem polynomial.card_roots_X_pow_sub_C {R : Type u} [comm_ring R] [is_domain R] {n : ℕ} (hn : 0 < n) (a : R) :

⇑multiset.card (polynomial.X ^ n - ⇑polynomial.C a).roots ≤ n

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theorem polynomial.le_root_multiplicity_map {K : Type u_1} {L : Type u_2} [comm_ring K] [comm_ring L] {p : polynomial K} {f : K →+* L} (hf : function.injective ⇑f) (a : K) :

polynomial.root_multiplicity a p ≤ polynomial.root_multiplicity (⇑f a) (polynomial.map f p)

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theorem polynomial.count_map_roots {K : Type u_1} {L : Type u_2} [comm_ring K] [is_domain K] [comm_ring L] {p : polynomial K} {f : K →+* L} (hf : function.injective ⇑f) (a : L) :

multiset.count a (multiset.map ⇑f p.roots) ≤ polynomial.root_multiplicity a (polynomial.map f p)

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theorem polynomial.roots_map_of_injective_card_eq_total_degree {K : Type u_1} {L : Type u_2} [comm_ring K] [is_domain K] [comm_ring L] [is_domain L] {p : polynomial K} {f : K →+* L} (hf : function.injective ⇑f) (hroots : ⇑multiset.card p.roots = p.nat_degree) :

multiset.map ⇑f p.roots = (polynomial.map f p).roots

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noncomputable def polynomial.nth_roots {R : Type u} [comm_ring R] [is_domain R] (n : ℕ) (a : R) :

multiset R

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

Equations

polynomial.nth_roots n a = (polynomial.X ^ n - ⇑polynomial.C a).roots

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@[simp]

theorem polynomial.mem_nth_roots {R : Type u} [comm_ring R] [is_domain R] {n : ℕ} (hn : 0 < n) {a x : R} :

x ∈ polynomial.nth_roots n a ↔ x ^ n = a

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@[simp]

theorem polynomial.nth_roots_zero {R : Type u} [comm_ring R] [is_domain R] (r : R) :

polynomial.nth_roots 0 r = 0

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theorem polynomial.card_nth_roots {R : Type u} [comm_ring R] [is_domain R] (n : ℕ) (a : R) :

⇑multiset.card (polynomial.nth_roots n a) ≤ n

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noncomputable def polynomial.nth_roots_finset (n : ℕ) (R : Type u_1) [comm_ring R] [is_domain R] :

finset R

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

Equations

polynomial.nth_roots_finset n R = (polynomial.nth_roots n 1).to_finset

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@[simp]

theorem polynomial.mem_nth_roots_finset {R : Type u} [comm_ring R] [is_domain R] {n : ℕ} (h : 0 < n) {x : R} :

x ∈ polynomial.nth_roots_finset n R ↔ x ^ n = 1

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@[simp]

theorem polynomial.nth_roots_finset_zero {R : Type u} [comm_ring R] [is_domain R] :

polynomial.nth_roots_finset 0 R = ∅

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theorem polynomial.coeff_comp_degree_mul_degree {R : Type u} [comm_ring R] [is_domain R] {p q : polynomial R} (hqd0 : q.nat_degree ≠ 0) :

(p.comp q).coeff ((p.nat_degree) * q.nat_degree) = (p.leading_coeff) * q.leading_coeff ^ p.nat_degree

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theorem polynomial.nat_degree_comp {R : Type u} [comm_ring R] [is_domain R] {p q : polynomial R} :

(p.comp q).nat_degree = (p.nat_degree) * q.nat_degree

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theorem polynomial.leading_coeff_comp {R : Type u} [comm_ring R] [is_domain R] {p q : polynomial R} (hq : q.nat_degree ≠ 0) :

(p.comp q).leading_coeff = (p.leading_coeff) * q.leading_coeff ^ p.nat_degree

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theorem polynomial.units_coeff_zero_smul {R : Type u} [comm_ring R] [is_domain R] (c : (polynomial R)ˣ) (p : polynomial R) :

↑c.coeff 0 • p = (↑c) * p

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@[simp]

theorem polynomial.nat_degree_coe_units {R : Type u} [comm_ring R] [is_domain R] (u : (polynomial R)ˣ) :

↑u.nat_degree = 0

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theorem polynomial.comp_eq_zero_iff {R : Type u} [comm_ring R] [is_domain R] {p q : polynomial R} :

p.comp q = 0 ↔ p = 0 ∨ polynomial.eval (q.coeff 0) p = 0 ∧ q = ⇑polynomial.C (q.coeff 0)

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theorem polynomial.zero_of_eval_zero {R : Type u} [comm_ring R] [is_domain R] [infinite R] (p : polynomial R) (h : ∀ (x : R), polynomial.eval x p = 0) :

p = 0

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theorem polynomial.funext {R : Type u} [comm_ring R] [is_domain R] [infinite R] {p q : polynomial R} (ext : ∀ (r : R), polynomial.eval r p = polynomial.eval r q) :

p = q

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def polynomial.root_set {T : Type w} [comm_ring T] (p : polynomial T) (S : Type u_1) [comm_ring S] [is_domain S] [algebra T 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

p.root_set S = ↑((polynomial.map (algebra_map T S) p).roots.to_finset)

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theorem polynomial.root_set_def {T : Type w} [comm_ring T] (p : polynomial T) (S : Type u_1) [comm_ring S] [is_domain S] [algebra T S] :

p.root_set S = ↑((polynomial.map (algebra_map T S) p).roots.to_finset)

source

@[simp]

theorem polynomial.root_set_zero {T : Type w} [comm_ring T] (S : Type u_1) [comm_ring S] [is_domain S] [algebra T S] :

0.root_set S = ∅

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@[simp]

theorem polynomial.root_set_C {S : Type v} {T : Type w} [comm_ring T] [comm_ring S] [is_domain S] [algebra T S] (a : T) :

(⇑polynomial.C a).root_set S = ∅

source

@[protected, instance]

noncomputable def polynomial.root_set_fintype {T : Type w} [comm_ring T] (p : polynomial T) (S : Type u_1) [comm_ring S] [is_domain S] [algebra T S] :

fintype ↥(p.root_set S)

Equations

p.root_set_fintype S = finset_coe.fintype (polynomial.map (algebra_map T S) p).roots.to_finset

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theorem polynomial.root_set_finite {T : Type w} [comm_ring T] (p : polynomial T) (S : Type u_1) [comm_ring S] [is_domain S] [algebra T S] :

(p.root_set S).finite

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theorem polynomial.is_unit_iff {R : Type u} [comm_ring R] [is_domain R] {f : polynomial R} :

is_unit f ↔ ∃ (r : R), is_unit r ∧ ⇑polynomial.C r = f

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theorem polynomial.coeff_coe_units_zero_ne_zero {R : Type u} [comm_ring R] [is_domain R] (u : (polynomial R)ˣ) :

↑u.coeff 0 ≠ 0

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theorem polynomial.degree_eq_degree_of_associated {R : Type u} [comm_ring R] [is_domain R] {p q : polynomial R} (h : associated p q) :

p.degree = q.degree

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theorem polynomial.degree_eq_one_of_irreducible_of_root {R : Type u} [comm_ring R] [is_domain R] {p : polynomial R} (hi : irreducible p) {x : R} (hx : p.is_root x) :

p.degree = 1

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theorem polynomial.leading_coeff_div_by_monic_of_monic {R : Type u} [comm_ring R] {p q : polynomial R} (hmonic : q.monic) (hdegree : q.degree ≤ p.degree) :

(p /ₘ q).leading_coeff = p.leading_coeff

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

source

theorem polynomial.leading_coeff_div_by_monic_X_sub_C {R : Type u} [comm_ring R] [is_domain R] (p : polynomial R) (hp : p.degree ≠ 0) (a : R) :

(p /ₘ (polynomial.X - ⇑polynomial.C a)).leading_coeff = p.leading_coeff

source

theorem polynomial.eq_of_monic_of_dvd_of_nat_degree_le {R : Type u} [comm_ring R] [is_domain R] {p q : polynomial R} (hp : p.monic) (hq : q.monic) (hdiv : p ∣ q) (hdeg : q.nat_degree ≤ p.nat_degree) :

q = p

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theorem polynomial.is_unit_of_is_unit_leading_coeff_of_is_unit_map {R : Type u} {S : Type v} [semiring R] [comm_ring S] [is_domain S] (φ : R →+* S) (f : polynomial R) (hf : is_unit f.leading_coeff) (H : is_unit (polynomial.map φ f)) :

is_unit f

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theorem polynomial.monic.irreducible_of_irreducible_map {R : Type u} {S : Type v} [comm_ring R] [is_domain R] [comm_ring S] [is_domain 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.

mathlib documentation

Theory of univariate polynomials #