Let $\Bbb F_{p^{n}}$ be the field of order $p^{n}$. Define a map $\phi: \Bbb F_{p^{n}}\to\Bbb F_{p^{n}}$ by $x \mapsto x^{p} -x$. My question is what is the order of im$(\phi)$?
I already know Frobenius automorphism $\sigma: \Bbb F_{p^{n}}\to\Bbb F_{p^{n}}$ via $x \mapsto x^{p}$ has order $n$. But it seems that any work with $\sigma$ does not involve $\phi$.
Any help would be much appreciated.