I failed to find an example of a field F with non-cyclic Group Aut(F).
Must Aut(F) be a Cyclic Group for any field F?
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Does $\mathbb R$ have a cyclic group of automorphisms? – Alessandro Apr 11 '21 at 16:43
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@Alessandro Yes, $Aut(\Bbb R)$ is trivial, see here. The trivial group is cyclic. But what about $Aut(\Bbb C)$? – Dietrich Burde Apr 11 '21 at 16:45
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2The simplest example is probably $\Bbb{Q}(\sqrt2,i)$. Then you might want to look at $Aut(\Bbb{C}(t)/\Bbb{C})=PGL_2(\Bbb{C})$. – reuns Apr 11 '21 at 17:14