Let $F$ be a field containing a $n$-th primitive root of unity, now given a finite subgroup $G$ of $F^*/(F^*)^n$, let $K$ be the field containing $F$ and $\alpha$ such that $\alpha^n (F^*)^n\in G$. How to show that kum$(K/F)\cong G$?
Notation: kum$(K/F)=\{ \alpha (F^*)^n: \alpha^n\in F^*\}$