$F$ is a field of characteristic $p$ and $a\neq c^p-c$ for $c\in F$. Then determine the galois group of $x^p-x-a$.
First I showed that this is an irreducible polynomial and has no multiple roots. This will imply that the polynomial had $p$- distinct roots in its splitting field. Hence we can get a $p$-cycle. This means that the Galois group is $S_p$. Is my reasoning correct?
Actually, if $\alpha$ is a root, then so is $\alpha+n$, for all $n\in\mathbb F_p$. Thus, adjoining one root gives the entire splitting field, and thus the splitting field is an extension of degree $p$. The Galois Group is thus isomorphic to $C_p$.