GCD structures on polynomials #

Definitions and basic results about polynomials over GCD domains, particularly their contents and primitive polynomials.

Main Definitions #

Let p : polynomial R.

p.content is the gcd of the coefficients of p.
p.is_primitive indicates that p.content = 1.

Main Results #

polynomial.content_mul: If p q : polynomial R, then (p * q).content = p.content * q.content.
polynomial.normalized_gcd_monoid: The polynomial ring of a GCD domain is itself a GCD domain.

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def polynomial.is_primitive {R : Type u_1} [comm_semiring R] (p : polynomial R) :

Prop

A polynomial is primitive when the only constant polynomials dividing it are units

Equations

p.is_primitive = ∀ (r : R), ⇑polynomial.C r ∣ p → is_unit r

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theorem polynomial.is_primitive_iff_is_unit_of_C_dvd {R : Type u_1} [comm_semiring R] {p : polynomial R} :

p.is_primitive ↔ ∀ (r : R), ⇑polynomial.C r ∣ p → is_unit r

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@[simp]

theorem polynomial.is_primitive_one {R : Type u_1} [comm_semiring R] :

1.is_primitive

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theorem polynomial.monic.is_primitive {R : Type u_1} [comm_semiring R] {p : polynomial R} (hp : p.monic) :

p.is_primitive

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theorem polynomial.is_primitive.ne_zero {R : Type u_1} [comm_semiring R] [nontrivial R] {p : polynomial R} (hp : p.is_primitive) :

p ≠ 0

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def polynomial.content {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] (p : polynomial R) :

p.content is the gcd of the coefficients of p.

Equations

p.content = p.support.gcd p.coeff

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theorem polynomial.content_dvd_coeff {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] {p : polynomial R} (n : ℕ) :

p.content ∣ p.coeff n

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@[simp]

theorem polynomial.content_C {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] {r : R} :

(⇑polynomial.C r).content = ⇑normalize r

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@[simp]

theorem polynomial.content_zero {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] :

0.content = 0

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@[simp]

theorem polynomial.content_one {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] :

1.content = 1

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theorem polynomial.content_X_mul {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] {p : polynomial R} :

(polynomial.X * p).content = p.content

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@[simp]

theorem polynomial.content_X_pow {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] {k : ℕ} :

(polynomial.X ^ k).content = 1

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@[simp]

theorem polynomial.content_X {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] :

polynomial.X.content = 1

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theorem polynomial.content_C_mul {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] (r : R) (p : polynomial R) :

((⇑polynomial.C r) * p).content = (⇑normalize r) * p.content

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@[simp]

theorem polynomial.content_monomial {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] {r : R} {k : ℕ} :

(⇑(polynomial.monomial k) r).content = ⇑normalize r

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theorem polynomial.content_eq_zero_iff {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] {p : polynomial R} :

p.content = 0 ↔ p = 0

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@[simp]

theorem polynomial.normalize_content {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] {p : polynomial R} :

⇑normalize p.content = p.content

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theorem polynomial.content_eq_gcd_range_of_lt {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] (p : polynomial R) (n : ℕ) (h : p.nat_degree < n) :

p.content = (finset.range n).gcd p.coeff

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theorem polynomial.content_eq_gcd_range_succ {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] (p : polynomial R) :

p.content = (finset.range p.nat_degree.succ).gcd p.coeff

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theorem polynomial.content_eq_gcd_leading_coeff_content_erase_lead {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] (p : polynomial R) :

p.content = gcd p.leading_coeff p.erase_lead.content

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theorem polynomial.dvd_content_iff_C_dvd {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] {p : polynomial R} {r : R} :

r ∣ p.content ↔ ⇑polynomial.C r ∣ p

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theorem polynomial.C_content_dvd {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] (p : polynomial R) :

⇑polynomial.C p.content ∣ p

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theorem polynomial.is_primitive_iff_content_eq_one {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] {p : polynomial R} :

p.is_primitive ↔ p.content = 1

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theorem polynomial.is_primitive.content_eq_one {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] {p : polynomial R} (hp : p.is_primitive) :

p.content = 1

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def polynomial.prim_part {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] (p : polynomial R) :

polynomial R

The primitive part of a polynomial p is the primitive polynomial gained by dividing p by p.content. If p = 0, then p.prim_part = 1.

Equations

p.prim_part = ite (p = 0) 1 (classical.some _)

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theorem polynomial.eq_C_content_mul_prim_part {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] (p : polynomial R) :

p = (⇑polynomial.C p.content) * p.prim_part

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@[simp]

theorem polynomial.prim_part_zero {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] :

0.prim_part = 1

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theorem polynomial.is_primitive_prim_part {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] (p : polynomial R) :

p.prim_part.is_primitive

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theorem polynomial.content_prim_part {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] (p : polynomial R) :

p.prim_part.content = 1

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theorem polynomial.prim_part_ne_zero {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] (p : polynomial R) :

p.prim_part ≠ 0

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theorem polynomial.nat_degree_prim_part {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] (p : polynomial R) :

p.prim_part.nat_degree = p.nat_degree

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@[simp]

theorem polynomial.is_primitive.prim_part_eq {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] {p : polynomial R} (hp : p.is_primitive) :

p.prim_part = p

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theorem polynomial.is_unit_prim_part_C {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] (r : R) :

is_unit (⇑polynomial.C r).prim_part

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theorem polynomial.prim_part_dvd {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] (p : polynomial R) :

p.prim_part ∣ p

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theorem polynomial.gcd_content_eq_of_dvd_sub {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] {a : R} {p q : polynomial R} (h : ⇑polynomial.C a ∣ p - q) :

gcd a p.content = gcd a q.content

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theorem polynomial.content_mul_aux {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] {p q : polynomial R} :

gcd (p * q).erase_lead.content p.leading_coeff = gcd ((p.erase_lead) * q).content p.leading_coeff

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@[simp]

theorem polynomial.content_mul {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] {p q : polynomial R} :

(p * q).content = (p.content) * q.content

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theorem polynomial.is_primitive.mul {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] {p q : polynomial R} (hp : p.is_primitive) (hq : q.is_primitive) :

(p * q).is_primitive

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@[simp]

theorem polynomial.prim_part_mul {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] {p q : polynomial R} (h0 : p * q ≠ 0) :

(p * q).prim_part = (p.prim_part) * q.prim_part

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theorem polynomial.is_primitive.is_primitive_of_dvd {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] {p q : polynomial R} (hp : p.is_primitive) (hdvd : q ∣ p) :

q.is_primitive

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theorem polynomial.is_primitive.dvd_prim_part_iff_dvd {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] {p q : polynomial R} (hp : p.is_primitive) (hq : q ≠ 0) :

p ∣ q.prim_part ↔ p ∣ q

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theorem polynomial.exists_primitive_lcm_of_is_primitive {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] {p q : polynomial R} (hp : p.is_primitive) (hq : q.is_primitive) :

∃ (r : polynomial R), r.is_primitive ∧ ∀ (s : polynomial R), p ∣ s ∧ q ∣ s ↔ r ∣ s

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theorem polynomial.dvd_iff_content_dvd_content_and_prim_part_dvd_prim_part {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] {p q : polynomial R} (hq : q ≠ 0) :

p ∣ q ↔ p.content ∣ q.content ∧ p.prim_part ∣ q.prim_part

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@[instance]

def polynomial.normalized_gcd_monoid {R : Type u_1} [comm_ring R] [integral_domain R] [normalized_gcd_monoid R] :

normalized_gcd_monoid (polynomial R)

Equations

polynomial.normalized_gcd_monoid = normalized_gcd_monoid_of_exists_lcm polynomial.normalized_gcd_monoid._proof_1