Let $F$ be a field with $\mathbb{char}(F) = p$. Prove that if $E/F$ is a finite extension with $p \nmid [E:F]$, then $E/F$ is a separable extension.
I've no idea how to prove this, I'm looking for some insight to lead me in the right direction.
I've considered a proof by contradiction, supposing $E/F$ is not separable so there is some $\alpha \in F$ such that the minimal polynomial of $\alpha$ is not separable, ie. $p(x) = (x - \alpha)^2g(x)$, but I can't see how to derive a contradiction from this.