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Prove that $n$ divides $\varphi(p^n-1)$.

I want to check if my proof is correct.

Proof. $\text{Gal}(\Bbb F_{p^n}/\Bbb F_p)$ is a cyclic group of order $n$ and $|\text{Aut}(\Bbb F_{p^n}^\times)| = \varphi(p^n-1)$. For each automorphism $\Bbb F_{p^n}^\times\to\Bbb F_{p^n}^\times$, we extend it to $\Bbb F_{p^n}\to\Bbb F_{p^n}$ by defining $0\mapsto 0$. Then $\text{Gal}(\Bbb F_{p^n}/\Bbb F_p)<\text{Aut}(\Bbb F_{p^n})$ so that $n\mid\varphi(p^n-1)$.

Is the extension make sense?

Sahan Manodya
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  • Your proof is correct. But it is, perhaps, unnecessarily high browed in the sense that we can do with theory simpler than the Galois theory of finite fields :-) – Jyrki Lahtonen Sep 13 '21 at 04:31
  • I think that you might actually add your argument as an answer to the latter duplicate target! – Jyrki Lahtonen Sep 13 '21 at 04:35
  • A remark on your logic. I think that it may be better to restrict field automorphisms to the multiplicative subgroup rather than extend automorphisms of the multiplicative group. That shows that the Galois group is a subgroup of the automorphism group of the cyclic multiplicative group, when Lagrange gives the desired conclusion. The reason for me having second thoughts is that when $\mathrm{Aut}(\Bbb{F}_{p^n})$ is mentioned, I would be inclined to assume that automorphisms of the additive group are intended rather than those of the multiplicative monoid. – Jyrki Lahtonen Sep 13 '21 at 04:40

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