Explicit Kummer theory for the rational numbers

Let G be a finitely generated multiplicative subgroup of as× having rank r. The ratio between nr and the Kummer degree [as(ζm,Gn): as(ζm)], where n divides m, is bounded independently of n and m. We prove that there exist integers m0,n0 such that the above ratio depends only on G, gcd(m,m0), and gcd(n,n0). Our results are very explicit and they yield an algorithm that provides formulas for all the above Kummer degrees (the formulas involve a finite case distinction).