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How do I prove that $x^p-x+a$ is irreducible in a field of $p$ elements when $a\neq 0$?
How to prove that $x^p - x - 1$ is irreducible over $F_{p}[x]$. I thought proving that this polynomial has no roots over $F_p$ but that is not enough to irreducibility.
In a splitting field $K$, if $\alpha\in K$ is a root, then $(\alpha+1)^p= \alpha^p+1$ implies that $\alpha+1$ is a root as well. We conclude that the cyclic group of order $p$ operates on the roots (by addition of multiples of $1$), and necessarily does so transitively (as there are $p$ roots). A non-trivial factor of the polynomial would correspond to a non-trivial subgroup, which does not exist.