Is there a field extension $K / \Bbb C$ such that $\mathrm{Aut}(K / \Bbb C)$ is a non-trivial abelian group?
Of course $K$ should be an infinite (non-algebraic) extension of $\Bbb C$, so it has a transcendental element $t \in K$. I know that $\mathrm{Aut}(\Bbb C(t) / \Bbb C) \cong PGL_2(\Bbb C)$. On the other hand, in Galois theory, if $K/F$ is abelian, then any subextension $L/F$ is abelian. But this doesn't apply to the subextension $\Bbb C(t) / \Bbb C$ of $K/\Bbb C$ since it is not algebraic. What can I do then?
(Side note: I'm not even sure that $K \supsetneq \Bbb C \implies \mathrm{Aut}(K / \Bbb C) \supsetneq \{1\}$.)
Thank you.