Let $K$ be a field containing all $n$-th roots of unity with char$K$ and $n$ coprime. An abelian galois extension $L$ (not necessarily finite) of $K$ is Kummerian if Gal$(K/L)$ has exponent $n \in \mathbb{N}$. There is an inclusion preserving bijection between Kummerian extensions $L$ of $K$ and subgroups $\Delta$ of $K^{\times}$ containing $(K^{\times})^n$ given by $$ L \longmapsto (L^{\times})^n \cap K^{\times} $$ $$ \Delta \longmapsto K(\Delta^{\frac{1}{n}}) $$ Thus, the largest Kummerian extension of $K$ is precisely $K((K^{\times})^{\frac{1}{n}})$. I want to determine its galois group.
I know that it is the limit of the galois groups of the intermediate galois extensions, and these galois groups are subgroups of $(\mathbb{Z}/n\mathbb{Z})^r$ for some $r$, but I would like to determine the group more precisely.
Thanks in advance.