In Lang's Algebra, I had hard times with this question:
Let $K$ be field with characteristic $p$ (a prime). Let $L|K$ be a finite extension of $K$, and suppose $gcd([L:K],p)=1$. Show $L$ is separable over $K$.
In order to have $L$ separable over $K$, one needs to have $K(\alpha)|K$ separable for each $\alpha \in L$.
We can say $L=K(\alpha_1,...,\alpha_m)$. It seems like I need to use the tower relation of these extensions and get some information about their degrees that will be relevant to the gcd thing. But I couldn't do these. A hint is welcomed.