# Problem regarding proving an extension of a field to be separable

The whole question looks like-

Let, $$x^p-x-1$$ be a polynomial over a field $$F$$ of characteristic $$p\ne 0$$ and $$\alpha$$ be a root of it. Prove that $$F(\alpha)$$ is separable extension over $$F$$.

I have tried a bit of the problem which goes as follows-
Note that, if $$F$$ is finite i.e. $$F\simeq\Bbb{Z}_p$$ then the result is obvious, since any irreducible polynomial over a finite field cannot have multiple root.
Again, $$f’(x)=-1\ne0$$, hence $$f(x)$$ has all roots simple.
Now, let $$p(x)\in F[x]$$ is the minimal polynomial of $$\alpha$$ over $$F$$. Then $$p(x)|f(x)\implies$$ any root of $$p(x)$$ is a root of $$f(x)\implies p(x)$$ cannot have multiple roots.
So, I get $$\alpha$$ is separable over $$F$$. From this I cannot get any idea how to show $$F(\alpha)$$ is separable over $$F$$.
Can anybody complete this proof? Thanks for assistance in advance.

• AN extension generated by separable elements is a separable extension. – Lord Shark the Unknown Dec 5 '18 at 12:54
• And how do you deal with the case $F = \mathbb{F}_p(t)$ ? To get the intuition for separability you probably need the concept of fixed field. Distinct roots implies we'll have enough automorphisms (of the splitting field) for $F$ being the fixed field of the Galois group. – reuns Dec 5 '18 at 12:58
• Lord Shark the Unknown, yes intuitively looks like. But I can't write the proof properly. – Biswarup Saha Dec 5 '18 at 13:07