The degree of Kummer extensions of number fields

Let K be a number field, and let α1,.,αr be elements of K× which generate a subgroup of K× of rank r. Consider the cyclotomic-Kummer extensions of K given by K(ζn,α1n1,.,αrnr), where ni divides n for all i. There is an integer x such that these extensions have maximal degree over K(ζg,α1g1,.,αrgr), where g =gcd(n,x) and gi =gcd(ni,x). We prove that the constant x is computable. This result reduces to finitely many cases the computation of the degrees of the extensions K(ζn,α1n1,.,αrnr) over K.