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Let $K$ be a fixed field in $\mathbb C$ (complex numbers) of an automorphism of $\mathbb C$. Prove that every finite extension of $K$ in $\mathbb C$ is cyclic.

Thank you for your help!

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    Do u mean field fixed by an automorphism of C?2011-05-03
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    Let $\sigma$ be the automorphism. Given an extension $L$ of $K$, first show that a power of $\sigma$ acts as identity on $L$. Now you have the following problem: let $L$ be a field, $G$ a finite subgroup of $\mathrm{Aut}(L)$. If $K$ is the subfield of $L$ fixed by $G$, then $L/K$ is Galois with Galois group $G$.2011-05-03
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    That's exactly what I mean.2011-05-03
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    Thank you Jiangwei! I will try that.2011-05-03
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    I might be confused about something but surely K is either $\mathbb R$ or $\mathbb C$? (So the group is either trivial or $C_2$)2011-05-03
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    I believe this only holds if you also want your automorphism to be continuous. You trivially have $\sigma(\mathbb{Q})=\mathbb{Q}$ and therefore by continuity $\sigma(\mathbb{R})=\mathbb{R}$. This is not necessarily the case if your automorphism is not continuous (at least I don't see how it follows just from the properties of an automorphism).2012-01-14

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