Automorphisms of splitting field of $x^p-x-a$ Let $p$ be a prime and consider the splitting field of $f(x) = x^{p} - x - a$ over $\mathbb{F}_{p}$.  I have worked out that the splitting field is $\mathbb{F}_{p}(\beta)$, where $\beta$ is a root of $f$.  Then the maps $\tau^{n}$, where $\tau: \beta \mapsto \beta + 1$ permute the roots, but I am having trouble seeing that they are actually automorphisms.  For example,
$$\tau(\beta + (\beta + k)) = \beta + (\beta + k) + 1,\\
\tau(\beta) + \tau(\beta + k) = \beta + 1 + (\beta + k) + 1.$$
What am I missing?

Your map sends $\beta \mapsto \beta + 1$ but not $x \mapsto x+1$ for all $x\in F_p(\beta)$ this is your mistake:
$$\tau(\beta+(\beta+k))\neq \beta+(\beta+k)+1.$$
For example if $k\in F_p$, $\tau(k)=k$ but $\tau(k)\neq k+1$.
