I'm wondering if anyone has classified all the fields $K$ such that $Gal(\bar{K}/K)$ is abelian?

The only examples I'm aware of are: finite fields, the real numbers $\mathbb{R}$ and $k((T))$ where $k$ is any algebraically closed field of characteristic 0. (One might also add any algebraically closed field as an example not listed above...)

Is there any other example of such fields?

An possibly easier question would be: can one apriori prove that $Gal(\bar{K}/K)$ is procyclic as suggested by the examples above?

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