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 \(\frac{p}{q}\) (written in lowest form) of the polynomial
\[ f(z) = a_n z^n + a_{n - 1} z^{n - 1} + \dots + a_1 z + a_0\]
has the property that \(q \mid a_n\), but \(a_n = 1\) by assumption and so \(q = \pm 1\). Thus \(\frac{p}{q} \in \mathbb{Z}\).