# How do I prove that x^p-x+ax^p-x+a is irreducible in a field with pp elements when a\neq 0a\neq 0?

Let $p$ be a prime. How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?

Right now I’m able to prove that it has no roots and that it is separable, but I have not a clue as to how to prove it is irreducible. Any ideas?

$x \to x^p$ is an automorphism sending $r$ to $r-a$ for any root $r$ of the polynomial. This operation is cyclic of order $p$, so that one can get from any root to any other by applying the automorphism several times. The Galois group thus acts transitively on the roots, which is equivalent to irreducibility.