Let $K$ be a field, $\bar{K}$ an algebraic closure and $\sigma \in \mathrm{Aut}_K \bar{K}$. Let $F=\{u \in \bar{K} \mid \sigma(u)=u\}$. Then prove $F$ is a field and every finite-dimensional extension of $F$ is cyclic.
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3Since you may have simply copied an exercise from a book, there is a risk of people down-voting your question or voting to close it for lack of thoughts of your own about it included in the question, indicating perhaps specifically where you got stuck when trying to solve it, or some question of your own about it. $\qquad$ – Michael Hardy Mar 04 '16 at 16:48
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There are two things to prove. Did you succeed in proving one of them and fail with the other? Proving $F$ is a field seems fairly easy. The other part I haven't thought about yet. $\qquad$ – Michael Hardy Mar 04 '16 at 16:54
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Do you know some Galois theory ? Being fixed by $\sigma$ is the same as being fixed by the subgroup $\langle \sigma\rangle$ it generates. – Captain Lama Mar 04 '16 at 19:52
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More answers to the same question. Still looking for a definite duplicate target. – Jyrki Lahtonen Mar 06 '16 at 13:59
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I deleted my answer because it was for all purposes a duplicate of this one. – Jyrki Lahtonen Mar 06 '16 at 14:12