mathlib documentation

ring_theory.polynomial.basic

Ring-theoretic supplement of data.polynomial. #

Main results #

@[instance]
def polynomial.char_p {R : Type u} [semiring R] (p : ) [h : char_p R p] :
def polynomial.degree_le (R : Type u) [comm_ring R] (n : with_bot ) :

The R-submodule of R[X] consisting of polynomials of degree ≤ n.

Equations
def polynomial.degree_lt (R : Type u) [comm_ring R] (n : ) :

The R-submodule of R[X] consisting of polynomials of degree < n.

Equations
theorem polynomial.mem_degree_le {R : Type u} [comm_ring R] {n : with_bot } {f : polynomial R} :
theorem polynomial.mem_degree_lt {R : Type u} [comm_ring R] {n : } {f : polynomial R} :
theorem polynomial.degree_lt_mono {R : Type u} [comm_ring R] {m n : } (H : m n) :
def polynomial.degree_lt_equiv (F : Type u_1) [field F] (n : ) :

The first n coefficients on degree_lt n form a linear equivalence with fin n → F.

Equations
def polynomial.frange {R : Type u} [comm_ring R] (p : polynomial R) :

The finset of nonzero coefficients of a polynomial.

Equations
theorem polynomial.frange_zero {R : Type u} [comm_ring R] :
theorem polynomial.mem_frange_iff {R : Type u} [comm_ring R] {p : polynomial R} {c : R} :
c p.frange ∃ (n : ) (H : n p.support), c = p.coeff n
theorem polynomial.frange_one {R : Type u} [comm_ring R] :
1.frange {1}
theorem polynomial.coeff_mem_frange {R : Type u} [comm_ring R] (p : polynomial R) (n : ) (h : p.coeff n 0) :

Given a polynomial, return the polynomial whose coefficients are in the ring closure of the original coefficients.

Equations
@[simp]
theorem polynomial.coeff_restriction {R : Type u} [comm_ring R] {p : polynomial R} {n : } :
@[simp]
theorem polynomial.coeff_restriction' {R : Type u} [comm_ring R] {p : polynomial R} {n : } :
@[simp]
@[simp]
theorem polynomial.degree_restriction {R : Type u} [comm_ring R] {p : polynomial R} :
@[simp]
theorem polynomial.monic_restriction {R : Type u} [comm_ring R] {p : polynomial R} :
@[simp]
theorem polynomial.restriction_zero {R : Type u} [comm_ring R] :
@[simp]
theorem polynomial.restriction_one {R : Type u} [comm_ring R] :
theorem polynomial.eval₂_restriction {R : Type u} [comm_ring R] {S : Type v} [ring S] {f : R →+* S} {x : S} {p : polynomial R} :
def polynomial.to_subring {R : Type u} [comm_ring R] (p : polynomial R) (T : subring R) (hp : (p.frange) T) :

Given a polynomial p and a subring T that contains the coefficients of p, return the corresponding polynomial whose coefficients are in `T.

Equations
@[simp]
theorem polynomial.coeff_to_subring {R : Type u} [comm_ring R] (p : polynomial R) (T : subring R) (hp : (p.frange) T) {n : } :
((p.to_subring T hp).coeff n) = p.coeff n
@[simp]
theorem polynomial.coeff_to_subring' {R : Type u} [comm_ring R] (p : polynomial R) (T : subring R) (hp : (p.frange) T) {n : } :
((p.to_subring T hp).coeff n).val = p.coeff n
@[simp]
theorem polynomial.support_to_subring {R : Type u} [comm_ring R] (p : polynomial R) (T : subring R) (hp : (p.frange) T) :
@[simp]
theorem polynomial.degree_to_subring {R : Type u} [comm_ring R] (p : polynomial R) (T : subring R) (hp : (p.frange) T) :
@[simp]
theorem polynomial.nat_degree_to_subring {R : Type u} [comm_ring R] (p : polynomial R) (T : subring R) (hp : (p.frange) T) :
@[simp]
theorem polynomial.monic_to_subring {R : Type u} [comm_ring R] (p : polynomial R) (T : subring R) (hp : (p.frange) T) :
@[simp]
theorem polynomial.to_subring_zero {R : Type u} [comm_ring R] (T : subring R) :
0.to_subring T _ = 0
@[simp]
theorem polynomial.to_subring_one {R : Type u} [comm_ring R] (T : subring R) :
1.to_subring T _ = 1
@[simp]
theorem polynomial.map_to_subring {R : Type u} [comm_ring R] (p : polynomial R) (T : subring R) (hp : (p.frange) T) :
def polynomial.of_subring {R : Type u} [comm_ring R] (T : subring R) (p : polynomial T) :

Given a polynomial whose coefficients are in some subring, return the corresponding polynomial whose coefficients are in the ambient ring.

Equations
theorem polynomial.coeff_of_subring {R : Type u} [comm_ring R] (T : subring R) (p : polynomial T) (n : ) :
@[simp]
theorem polynomial.frange_of_subring {R : Type u} [comm_ring R] (T : subring R) {p : polynomial T} :
theorem ideal.polynomial_mem_ideal_of_coeff_mem_ideal {R : Type u} [comm_ring R] (I : ideal (polynomial R)) (p : polynomial R) (hp : ∀ (n : ), p.coeff n ideal.comap polynomial.C I) :
p I

If every coefficient of a polynomial is in an ideal I, then so is the polynomial itself

theorem ideal.mem_map_C_iff {R : Type u} [comm_ring R] {I : ideal R} {f : polynomial R} :
f ideal.map polynomial.C I ∀ (n : ), f.coeff n I

The push-forward of an ideal I of R to polynomial R via inclusion is exactly the set of polynomials whose coefficients are in I

theorem ideal.quotient_map_C_eq_zero {R : Type u} [comm_ring R] {I : ideal R} (a : R) (H : a I) :

If I is an ideal of R, then the ring polynomials over the quotient ring I.quotient is isomorphic to the quotient of polynomial R by the ideal map C I, where map C I contains exactly the polynomials whose coefficients all lie in I

Equations

If P is a prime ideal of R, then R[x]/(P) is an integral domain.

theorem ideal.is_prime_map_C_of_is_prime {R : Type u} [comm_ring R] {P : ideal R} (H : P.is_prime) :

If P is a prime ideal of R, then P.R[x] is a prime ideal of R[x].

Given any ring R and an ideal I of polynomial R, we get a map R → R[x] → R[x]/I. If we let R be the image of R in R[x]/I then we also have a map R[x] → R'[x]. In particular we can map I across this map, to get I' and a new map R' → R'[x] → R'[x]/I. This theorem shows I' will not contain any non-zero constant polynomials

polynomial R is never a field for any ring R.

theorem ideal.eq_zero_of_constant_mem_of_maximal {R : Type u} [comm_ring R] (hR : is_field R) (I : ideal (polynomial R)) [hI : I.is_maximal] (x : R) (hx : polynomial.C x I) :
x = 0

The only constant in a maximal ideal over a field is 0.

def ideal.of_polynomial {R : Type u} [comm_ring R] (I : ideal (polynomial R)) :

Transport an ideal of R[X] to an R-submodule of R[X].

Equations
theorem ideal.mem_of_polynomial {R : Type u} [comm_ring R] {I : ideal (polynomial R)} (x : polynomial R) :
def ideal.degree_le {R : Type u} [comm_ring R] (I : ideal (polynomial R)) (n : with_bot ) :

Given an ideal I of R[X], make the R-submodule of I consisting of polynomials of degree ≤ n.

Equations
def ideal.leading_coeff_nth {R : Type u} [comm_ring R] (I : ideal (polynomial R)) (n : ) :

Given an ideal I of R[X], make the ideal in R of leading coefficients of polynomials in I with degree ≤ n.

Equations
theorem ideal.mem_leading_coeff_nth {R : Type u} [comm_ring R] (I : ideal (polynomial R)) (n : ) (x : R) :
x I.leading_coeff_nth n ∃ (p : polynomial R) (H : p I), p.degree n p.leading_coeff = x
theorem ideal.leading_coeff_nth_mono {R : Type u} [comm_ring R] (I : ideal (polynomial R)) {m n : } (H : m n) :
def ideal.leading_coeff {R : Type u} [comm_ring R] (I : ideal (polynomial R)) :

Given an ideal I in R[X], make the ideal in R of the leading coefficients in I.

Equations
theorem ideal.mem_leading_coeff {R : Type u} [comm_ring R] (I : ideal (polynomial R)) (x : R) :
x I.leading_coeff ∃ (p : polynomial R) (H : p I), p.leading_coeff = x
theorem ideal.is_fg_degree_le {R : Type u} [comm_ring R] (I : ideal (polynomial R)) [is_noetherian_ring R] (n : ) :
@[instance]

Hilbert basis theorem: a polynomial ring over a noetherian ring is a noetherian ring.

theorem polynomial.exists_irreducible_of_degree_pos {R : Type u} [integral_domain R] [wf_dvd_monoid R] {f : polynomial R} (hf : 0 < f.degree) :
∃ (g : polynomial R), irreducible g g f
theorem polynomial.linear_independent_powers_iff_aeval {R : Type u} {M : Type w} [comm_ring R] [add_comm_group M] [module R M] (f : M →ₗ[R] M) (v : M) :
linear_independent R (λ (n : ), (f ^ n) v) ∀ (p : polynomial R), ((polynomial.aeval f) p) v = 0p = 0
theorem polynomial.disjoint_ker_aeval_of_coprime {R : Type u} {M : Type w} [comm_ring R] [add_comm_group M] [module R M] (f : M →ₗ[R] M) {p q : polynomial R} (hpq : is_coprime p q) :
theorem polynomial.sup_aeval_range_eq_top_of_coprime {R : Type u} {M : Type w} [comm_ring R] [add_comm_group M] [module R M] (f : M →ₗ[R] M) {p q : polynomial R} (hpq : is_coprime p q) :
theorem polynomial.sup_ker_aeval_le_ker_aeval_mul {R : Type u} {M : Type w} [comm_ring R] [add_comm_group M] [module R M] {f : M →ₗ[R] M} {p q : polynomial R} :
theorem polynomial.sup_ker_aeval_eq_ker_aeval_mul_of_coprime {R : Type u} {M : Type w} [comm_ring R] [add_comm_group M] [module R M] (f : M →ₗ[R] M) {p q : polynomial R} (hpq : is_coprime p q) :
@[instance]

The multivariate polynomial ring in finitely many variables over a noetherian ring is itself a noetherian ring.

Auxiliary lemma: Multivariate polynomials over an integral domain with variables indexed by fin n form an integral domain. This fact is proven inductively, and then used to prove the general case without any finiteness hypotheses. See mv_polynomial.integral_domain for the general case.

Auxiliary definition: Multivariate polynomials in finitely many variables over an integral domain form an integral domain. This fact is proven by transport of structure from the mv_polynomial.integral_domain_fin, and then used to prove the general case without finiteness hypotheses. See mv_polynomial.integral_domain for the general case.

Equations
theorem mv_polynomial.eq_zero_or_eq_zero_of_mul_eq_zero {R : Type u} [integral_domain R] {σ : Type v} (p q : mv_polynomial σ R) (h : p * q = 0) :
p = 0 q = 0
theorem mv_polynomial.map_mv_polynomial_eq_eval₂ {R : Type u} {σ : Type v} [comm_ring R] {S : Type u_1} [comm_ring S] [fintype σ] (ϕ : mv_polynomial σ R →+* S) (p : mv_polynomial σ R) :
theorem mv_polynomial.quotient_map_C_eq_zero {R : Type u} {σ : Type v} [comm_ring R] {I : ideal R} {i : R} (hi : i I) :
theorem mv_polynomial.mem_ideal_of_coeff_mem_ideal {R : Type u} {σ : Type v} [comm_ring R] (I : ideal (mv_polynomial σ R)) (p : mv_polynomial σ R) (hcoe : ∀ (m : σ →₀ ), mv_polynomial.coeff m p ideal.comap mv_polynomial.C I) :
p I

If every coefficient of a polynomial is in an ideal I, then so is the polynomial itself, multivariate version.

theorem mv_polynomial.mem_map_C_iff {R : Type u} {σ : Type v} [comm_ring R] {I : ideal R} {f : mv_polynomial σ R} :

The push-forward of an ideal I of R to mv_polynomial σ R via inclusion is exactly the set of polynomials whose coefficients are in I