## f=x^p-x-a in a field of characteritic p

$\mathbf{Problem:}$ Let $F$ be of characteritic $p$ and let $a\in F$. Show that $f(x)=x^p-x-a$ has no multiple roots and that $f(x)$ is irreducible in $F[x]$ if and only if $a\neq c^p-c$ for any $c\in F$.
$\mathbf{Proof:}$ $f'(x)=-1$, so $(f(x),f'(x))=1$ which means $f(x)$ is separable.
For the second part. If $f$ is irreducible then immediately it has no roots in $F$. Conversely, suppose $c$ is a root of $f$ in some splitting field, $c$ is not in $F$. Then by Freshman’s dream, one can verify $c+1,c+2,\cdots, c+p-1$ are also roots of $f$.

$\displaystyle f(x)=(x-c)(x-c-1)\cdots(x-p+1)$

The irreducible polynomial $g(x)$ of $c$ is a factor of $f(x)$. $g(x)$ must contain all the roots of $f(x)$, otherwise the sum of these roots will results in $c\in F$. So $f(x)=g(x)$ is irreducible.
$\mathbf{Remark:}$ Jacoboson, Algebra I, p234.