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Mathlib.RingTheory.Polynomial.RationalRoot

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 : A[X] in A's field of fractions are of the form x / y with x y : A, x ∣ p.coeff 0 and y ∣ p.leadingCoeff. The corollary is the integral root theorem isInteger_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 #

theorem scaleRoots_aeval_eq_zero_of_aeval_mk'_eq_zero {A : Type u_1} {S : Type u_4} [CommRing A] [CommRing S] {M : Submonoid A} [Algebra A S] [IsLocalization M S] {p : Polynomial A} {r : A} {s : { x // x M }} (hr : ↑(Polynomial.aeval (IsLocalization.mk' S r s)) p = 0) :
↑(Polynomial.aeval (↑(algebraMap A S) r)) (Polynomial.scaleRoots p s) = 0
theorem num_dvd_of_is_root {A : Type u_1} {K : Type u_2} [CommRing A] [IsDomain A] [UniqueFactorizationMonoid A] [Field K] [Algebra A K] [IsFractionRing A K] {p : Polynomial A} {r : K} (hr : ↑(Polynomial.aeval r) p = 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 den_dvd_of_is_root {A : Type u_1} {K : Type u_2} [CommRing A] [IsDomain A] [UniqueFactorizationMonoid A] [Field K] [Algebra A K] [IsFractionRing A K] {p : Polynomial A} {r : K} (hr : ↑(Polynomial.aeval r) p = 0) :

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

theorem isInteger_of_is_root_of_monic {A : Type u_1} {K : Type u_2} [CommRing A] [IsDomain A] [UniqueFactorizationMonoid A] [Field K] [Algebra A K] [IsFractionRing A K] {p : Polynomial A} (hp : Polynomial.Monic p) {r : K} (hr : ↑(Polynomial.aeval r) p = 0) :

Integral root theorem: if r : f.codomain is a root of a monic polynomial over the ufd A, then r is an integer