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ring_theory.polynomial.rational_root

Rational root theorem and integral root theorem

This file contains the rational root theorem and integral root theorem. The rational root theorem for a unique factorization domain A with localization S, states that the roots of p : polynomial A in A's field of fractions are of the form x / y with x y : A, x ∣ p.coeff 0 and y ∣ p.leading_coeff. The corollary is the integral root theorem is_integer_of_is_root_of_monic: if p is monic, its roots must be integers. Finally, we use this to show unique factorization domains are integrally closed.

References

• https://en.wikipedia.org/wiki/Rational_root_theorem
theorem scale_roots_aeval_eq_zero_of_aeval_mk'_eq_zero {A : Type u_1} {S : Type u_4} [comm_ring S] {M : submonoid A} {f : S} {p : polynomial A} {r : A} {s : M} :
(polynomial.aeval (f.mk' r s)) p = 0(polynomial.aeval ((f.to_map) r)) s) = 0

theorem num_is_root_scale_roots_of_aeval_eq_zero {A : Type u_1} {K : Type u_2} [field K] (g : K) {p : polynomial A}  :
p = 0 (g.denom x)).is_root (g.num x)

theorem num_dvd_of_is_root {A : Type u_1} {K : Type u_2} [field K] {f : K} {p : polynomial A}  :
p = 0f.num r p.coeff 0

Rational root theorem part 1: if r : f.codomain is a root of a polynomial over the ufd A, then the numerator of r divides the constant coefficient

theorem denom_dvd_of_is_root {A : Type u_1} {K : Type u_2} [field K] {f : K} {p : polynomial A}  :
Rational root theorem part 2: if r : f.codomain is a root of a polynomial over the ufd A, then the denominator of r divides the leading coefficient
Integral root theorem: if r : f.codomain is a root of a monic polynomial over the ufd A, then r is an integer