Let $K$ be a field of characteristic $p>0$ and let $L$ be the splitting field of the polynomial $f(t)=t^{p}-t+a$ over $K$, where $a \in K$. Let $\alpha \in L$ be a root of $f(t)$.

If $L \neq K$, show that $f(t)$ is irreducible over $K$, that $L=K(\alpha)$, and that $L$ is a Galois extension of $K$. What is $\operatorname{Gal}(L / K)$ ?