Rational number in $\mathbb{Z}[\omega]$ should be integer.

Question

Let $\omega = \cos \frac{2\pi}{p} + i \sin \frac{2\pi}{p}$ for some prime number $p > 2$. Then how to prove that if $q \in \mathbb{Q} \cap \mathbb{Z}[\omega]$, $q$ must be integer.

Answer 1

All elements of $\mathbb{Z}[\omega]$ are algebraic integers, meaning that they are solutions of monic polynomials over $\mathbb{Z}$. But then Gauss's lemma says that all rational solutions of monic polynomials over the integers are in fact integers.