I have to prove that given $F \subset E$, where $F$ is a field of characteristic p and supposing that $p(x)=x^p-x-a$ is irreducible in $F[X]$, with $\alpha \in E$ a root of $p(x)$. Then is $F(\alpha)/F$ a Galois extension.
I'm trying to see that this is a normal and separable extension.
So far I know that, given the characteristic of the field, the polynomial $f(x)=x^p−x$ has the property:
With $x_1$ and $x_2$ being two elements of an extension field of $F$. And because of little Fermat we know $f(k)=k^p−k=0$ for all $k∈F$. So, if $\alpha$ is a root of $p(x)=x^p−x-a$, then:
so all the elements $r+k$ with $k∈F$ are roots of $p(x)$, and as there are $p$ of them. So we've shown that there exists an irreducible polynomial whose roots, together with the elements of $F$, generate $F(\alpha)$ (clearly), and therefore it is a normal extension. But how can I prove it is separable? Or is there a better way to prove this is a Galois extension? Thank you, for any help.