Rational Roots of Monic Integer Polynomials

Theorem

The only rational roots of a monic integer polynomial are integers.

Proof

This is a direct consequence of the rational root lemma. In particular, any root pq (written in lowest form) of the polynomial

f(z)=anzn+an1zn1++a1z+a0

has the property that qan, but an=1 by assumption and so q=±1. Thus pqZ.