I'm taking an intro to Galois theory course, which is rather exciting. We had the following question in a practice paper:
Let $K$ be a field of characteristic $p$, and let $L/K$ be a finite extension with $\left[L:K\right]$ prime to $p$. Show that $L/K$ is separable.
Note that I took "$\left[L:K\right]$ prime to $p$" to mean $p\nmid\left[L:K\right]$.
I managed to reduce the problem to a seemingly simpler one. $L/K$ is separable if and only if $K\left(\alpha\right)$ is separable for all $\alpha\in L$. Also, from the multiplication rule, $p\nmid\left[K\left(\alpha\right):K\right]$. So it's sufficent to prove the proposition for $L=K\left(\alpha\right)$.
I couldn't proceed beyond that. I tried to play a bit with the general form of $\mathrm{irr}\left(\alpha,K\right)$, but that didn't inspire any solution. I had hoped that I might be able to show that $\gcd\left(f,f'\right)=1$, but was unsuccessful. I then tried to understand how an embedding of $L/K$ would "work", but that didn't lead me anywhere either.
I got the sense I'm missing something rather basic. I'm still a newbie when it comes to finite fields and fields with non-zero characteristic. They were mentioned in previous courses I took (say, linear algebra), but we didn't use them as much as fields of zero-characteristic.
I'd love a nudge in the right direction. Thanks!