The degree of Kummer extensions of number fields

Abstract

Let [Formula: see text] be a number field, and let [Formula: see text] be elements of [Formula: see text] which generate a subgroup of [Formula: see text] of rank [Formula: see text]. Consider the cyclotomic-Kummer extensions of [Formula: see text] given by [Formula: see text], where [Formula: see text] divides [Formula: see text] for all [Formula: see text]. There is an integer [Formula: see text] such that these extensions have maximal degree over [Formula: see text], where [Formula: see text] and [Formula: see text]. We prove that the constant [Formula: see text] is computable. This result reduces to finitely many cases the computation of the degrees of the extensions [Formula: see text] over [Formula: see text].