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?



Classify all the fields with abelian absolute Galois group
Geyer in [Unendliche algebraische Zahlkörper, über denen jede Gleichung auflösbar von beschränkter Stufe ist](https://doi.org/10.1016/0022-314X(69)90050-X), Satz 1.13 and the paragraph after that, gives a full characterization of which abelian profinite groups occur as absolute Galois groups: They are either $\mathbb{Z}/2\mathbb{Z}$ or $\prod\_p\mathbb{Z}\_p^{c(p)}$ for some cardinal numbers $c(p)$.


Side remark: For *algebraic* extensions of $\mathbb{Q}$, abelian absolute Galois groups are in fact procyclic, see Satz 2.3 in the same paper.


