Kummer theory for number fields via entanglement groups

Abstract

Let K be a number field, and let G be a finitely generated subgroup of K×. We are interested in computing the degree of the cyclotomic-Kummer extension K(Gn) over K, where Gn consists of all n-th roots of the elements of G. We develop the theory of entanglements introduced by Lenstra, and we apply it to compute the above degrees.