If $K$ has only finitely many Galois extensions, then $K$ must be either separably closed or real closed. Are there any other fields whose abelianizations are finite extensions (i.e. whose absolute Galois groups have finite abelianizations)?



Is there a field with finitely many abelian extensions, that is neither separably closed nor real closed?
Here is another example, which also answers the "followup question":
Let $K$ be the field of Laurent series over $\mathbb{R}$. Its absolute Galois group is the infinite profinite dihedral group $\hat{\mathbb{Z}}\rtimes(\mathbb{Z}/2\mathbb{Z})$, where the action is by inversion. This group is the free profinite product of two groups of order $2$. Its abelianization is therefore $(\mathbb{Z}/2\mathbb{Z})^2$. In particular, $K$ has the desired property.


