Divisibility Parameters and the Degree of Kummer Extensions of Number Fields

Abstract

Abstract Let K be a number field, and let ℓ be a prime number. Fix some elements α1,...,αr of K× which generate a subgroup of K× of rank r. Let n1,...,nr, m be positive integers with m ⩾ ni 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(ζℓm, α1ℓn1,…,αrℓnr \root {{\ell ^{{n_1}}}} \of {{\alpha _1}} , \ldots ,\root {{\ell ^{{n_r}}}} \of {{\alpha _r}} ) over K(ζℓm) for all n1,..., nr, 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.