I'm hoping someone can give me a nudge in the right direction...
Let $F$ be a finite field, and let $f(x)$ be a nonconstant polynomial whose derivative is the zero polynomial. Prove that $f$ cannot be irreducible over $F$.
I've got that every root of $f$ is a multiple root and that for $F=\mathbb{F}_{p^r}$, the exponent of every term of $f$ is a multiple of $p$.