Let $F$ be a field of characteristic $p$ and $a \in F$ not a $p$th power. Then the splitting field of $f = X^p - a \in F[X]$ has only one root of $f$. Thus when considering $|\text{Aut}(E/F)| = [E:F]$ it's important that $f$ be separable.
Please help me understand this. What is required to prove the statement in the title?