$ p \in Q[x] $ has as a root a fifth primitive root of unity, then every fifth primitive root of unity is a root of $p$. I'm extremely stuck. Can't figure it. 
The conjugate is easy: let $w$ be a primitive root of unity, then $w^{-1}$ will also be a root, that's easy. But I'm missing $w^2$ and $w^3$. Why would they be also roots of p? 

Because $x^{5} - 1 = (x - 1) (x^{4} + x^{3} + x^{2} + x + 1)$, $w$ will be a root of f = $x^{4} + x^{3} + x^{2} + x + 1$. I think there is no alternative to proving that $f$ is irreducible in $\mathbb{Q}[x]$, which can be done with Eisenstein's criterion.
