Originally Posted by

**Sampras** We know that $\displaystyle \Phi_{p}(x) = \frac{x^p-1}{x-1} = x^{p-1}+x^{p-2}+ \dots + x+1 $ is irreducible over $\displaystyle \mathbb{Q} $ for every prime $\displaystyle p $. Suppose $\displaystyle \gamma $ is a zero of $\displaystyle \Phi_{p}(x) $ and consider the field $\displaystyle \mathbb{Q}(\gamma) $.

(a) Show that $\displaystyle \gamma, \gamma^2, \dots, \gamma^{p-1} $ are distinct zeros of $\displaystyle \Phi_{p}(x) $ and show that they are all the zeros of $\displaystyle \Phi_{p}(x) $.