Kummer theory for p-adic fields

Let K be a p-adic field, namely a finite extension of the field of p-adic numbers Qₚ. If G is a finitely generated subgroup of K^×, we describe how to compute the degrees of the Kummer extensions K(ζₙ, √[n]G)/K(ζₙ) for all positive integers n. Moreover, we compare a~Kummer extension of number fields with the corresponding Kummer extensions of p-adic fields which arise by completion.