= Solution
In characteristic $p$, the binomial theorem gives $(\alpha+1)^p=\alpha^p+1$. Hence
$$
(\alpha+1)^p-(\alpha+1)-a=\alpha^p-\alpha-a=0.
$$
Iterating shows that all $\alpha+j$, with $j\in\mathbb F_p$, are roots. They are distinct, since the prime-<field> elements $j$ are distinct. There are $p$ of them, equal to the <polynomial>'s degree. Thus if $\alpha\in F$,
$$
\boxed{X^p-X-a=\prod_{j\in\mathbb F_p}(X-\alpha-j)}
$$
splits into distinct linear factors in $F[X]$. Equivalently, its derivative is $-1$, so no repeated roots are possible.
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