Prove that $f(x)$ divides $x^{p^n} - x$ if and only if $d := \deg f(x)$ divides $n$.
I believe that I have the backward direction covered: Let $d \mid n$ say $n = dq$ for some $q$ in $\mathbb{F}_p[x]$. Consider the field $\mathbb{F}_p[x]/(f(x))$ which has $p^d$ elements. Take an element $x+I$ from the field (here $I = (f(x))$) so we have: $(x+I)^{p^n} = (x+I)^{p^{dq}}$. As long as you keep factoring out $(x+I)$ with the $p^d$ power you will get $(x+I)$ so $x^{p^n} - x \in (f(x))$.
I am having trouble getting to the other direction.