Let $\mathbb F_p$ be a field with $p$ elements and consider f$f(x)\in \mathbb F_p[X]$ irreducible of degree $n$. Then a splitting field for $f(x)$ has $p^n$ elements.
Write $S_f$ for the splitting field of $f(x)$. Basically I have to show that $[S_f:\mathbb F_p]=n$ and the conclusion will follow. But I can't find a way to show that. I already deduced that $f(x)$ is separable and thus has $n$ different roots in $S_f$, but it does not seem helpful.
Any ideas? Thanks!