We have $GF(p)$ where $p$ is some prime. The polynomial $f(x)$ is irreducible over $GF(p)$. Show that $f(x)$ divides $g(x) = x^{p^{n}}-x \in GF(p)[x]$ if and only if deg$(f(x))$ divides $n$.
I assume that $f(x)$ divides $g(x)$. Then $g(x) = q(x)f(x)$ for some polynomial $q(x)$, and deg$(g(x))$ = deg$(q(x))$deg$(f(x))$. So the degrees of $q$ and $f$ must be some powers of $p$, say $s$ and $r$ respectively, so we have $p^n = p^{s}p^{r}$. This is where I'm stuck. I would appreciate some hints so I can move forward, but please don't post entire solution.