Show that if $\alpha$ is a root of $X^p-X-a$ in a characteristic $p>0$ field $k$.Then $L=k(\alpha)$ is normal in $k$.

Question

I am stuck with the following problem in Galois theory regarding characteristic $p$ fields.The problem is as follows:

Let $k$ be a characteristic $p>0$ field and $a\in k$ and $f(X)=X^p-X-a\in k[X]$.Prove that if $\alpha$ is a root of $f(X)$ in an algebraic closure of $k$,then $L=k(\alpha)$ is a normal extension of $k$.

I want some idea so that I can solve it.What is it that I might be missing?Is there some trick with the characteristic $p$ which I am overlooking?Can someone please give me some hint?

My attempt:

I was trying to construct the other roots from $\alpha$.But I cannot find how to do that.