irrational solutions of polynomial

Question

How to prove that if $x$ satisfies

$x^n+a_{n-1}x^{n-1}+\dots+a_0 = 0$

for some integers $a_{n-1}$, ..., $a_0$, then $x$ is irrational unless x is an integer.

Thanks for your help in advance.

Answer 1

Suppose $x=\frac p q$ is a root where $p$ and $q$ are integers, $q >0$ and $(p,q)=1$. Then $p^{n}+a_{n-1} p^{n-1}q+\cdots+a_oq^{n}=0$. This implies that $q$ divides $p^{n}$. But $(p,q)=1$ so $q$ must be $1$. So $x$ is an integer.