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Let $F$ be a finite field and let $K/F$ be a field extension of degree 6. Then the Galois group of $K/F$ is isomorphic to $S_3$ or $C_6$?

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    $\begingroup$ What do you know already about extensions of finite fields? E.g. that they are cyclic? $\endgroup$ – Torsten Schoeneberg Jun 12 '18 at 14:09
  • $\begingroup$ @TorstenSchoeneberg oh yes.. surely. I removed other two options i.e. $S_6$ and ${e}$ and was stuck here. $\endgroup$ – ChakSayantan Jun 12 '18 at 14:42
  • $\begingroup$ @ChakSayantan is $S_3$ cyclic? $\endgroup$ – Kenny Lau Jun 12 '18 at 23:53
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We have the general result that $\text{Gal}(\Bbb{F}_{p^n}/\Bbb{F}_p) \cong \Bbb{Z}_n$. This follows from the existence of the Frobenius automorphism $\sigma : \Bbb{F}_{p^n} \rightarrow \Bbb{F}_{p^n}$ given by $\sigma(\alpha)=\alpha^p$ for $\alpha \in \Bbb{F}_{p^n}$. So in your case, since $[K:F]=6$, the Galois group will be isomorphic to the cyclic group of order $6$.

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