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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.

user
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    The kernel of $\phi$ is the subfield $\Bbb{F}_p$, and its image consists of the elements of trace zero. – Jyrki Lahtonen Apr 19 '16 at 22:51
  • My question is answered at http://math.stackexchange.com/questions/1384817/proof-check-number-of-elements-of-mathbbf-pn-of-the-form-ap-a-fo?rq=1. Plz close this post as it duplicated. – user Apr 20 '16 at 05:43

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