Linear relations among radicals

Let $K$ be a field, fix an algebraic closure $\bar{K}$, and let $G$ be a subgroup of ${\bar{K}}^{\times }$. We are able to give a closed formula for the ratio between the degree $[K(G):K]$ and the index $\mathrm{\mid }G{K}^{\times }:K^{\times }\mathrm{\mid }$, provided that the latter is finite. Our formula explains all the $K$-linear relations among radicals, which (beyond the ones stemming from the multiplicative group $G{K}^{\times }\mathrm{/}{K}^{\times }$) are generated by relations among roots of unity and single radicals. Our work builds on results by Rybowicz, which in turn are based on work by Kneser and Schinzel.