Show that $\displaystyle \sqrt {2} \in \mathbb {Q} [ \omega ] $, where $\displaystyle \omega $ is a primitive 8th root of unity.

Proof. Now $\displaystyle \omega = e^ {2 \pi i /8} = cos (2 \pi /8 ) + i sin (2 \pi /8 ) $. So I have $\displaystyle \omega ^{8} = 1 $