Divisibility Parameters and the Degree of Kummer Extensions of Number Fields

Let K be a number field, and let ℓ be a prime number. Fix some elements α₁,...,α_r of K^× which generate a subgroup of K^× of rank r. Let n₁,...,n_r, m be positive integers with m ⩾ n_i for every i. We show that there exist computable parametric formulas (involving only a finite case distinction) to express the degree of the Kummer extension K(ζ_ℓ^m, α1ℓn1,…,αrℓnr \root {{\ell ^{{n_1}}}} \of {{\alpha _1}} , \ldots ,\root {{\ell ^{{n_r}}}} \of {{\alpha _r}} ) over K(ζ_ℓ^m) for all n₁,..., n_r, m. This is achieved with a new method with respect to a previous work, namely we determine explicit formulas for the divisibility parameters which come into play.