Given is the polynomial $f=X^p-X+a \in K[X]$, where $K$ is a field of characteristic $p$ and $a \in K$.
My task now is to show that every simple extension of $f$ is already the splitting field of $f$.
I should already show, that if $f$ has no root in $K$, it is irreducible.
My professor gave the hint to make use of $f$'s seperability and the fact that $f(X)=f(X+1)$. I've already proven this by applying $f$ to $X+1$ and by using the criterium for seperabilty by which it is equivalent to show that $gcd(f,f')=1$, where $f'$ is the formal deviation of $f$.