Question: I want to show that every algebraic extension of a finite field $F$ is separable.
Thoughts: I think I know how to do this: let $F\subseteq E$ be an algebraic extension of $F$. Let $a\in E$. Then there is some $f(x)\in F[x]$ irreducible such that $f(a)=0$. So, $f(x)=c_nx^n+\dots +c_1x+c_0$ where $c_i\in F$, $n\in\mathbb{N}$, $c_n\neq0$. So, $f'(x)=nc_nx^{n-1}+\dots +c_1$. Then we consider if $f'(x)=0$ then $(f(x),f'(x))=1$ so $f(x)$ is serpable, and if $f'(x)\neq0$ then I'm going through and eventually getting a contradiction to $f(x)$ being irreducible, thus finishing my proof. However, it takes about a half of a page to get that contradiction and I was wondering if there was maybe a shorter, or less computational, way of proving this. Thank you so much!