I'm trying to prove that every finite extension of a finite field is separable. I found a solution on internet which says:
Let $F$ be a finite field and $E$ be an extension of $F$ having $p^n$ elements. Then $E=F(\alpha)$, where $\alpha \in E$ and so $\alpha^{p^n} -\alpha=0$. This implies $\alpha$ is a separable element, and hence $F(\alpha)$ is a separable extension of F.
I don't understand why $\alpha^{p^n} -\alpha=0$ and why $\alpha$ is a separable element, I need help.
Thanks