Kummer Theory for Multiquadratic or Quartic Cyclic Number Fields

Let K be a number field which is multiquadratic or quartic cyclic. We prove several results about the Kummer extensions of K, namely concerning the intersection between the Kummer extensions and the cyclotomic extensions of K. For G a finitely generated subgroup of K^×, we consider the cyclotomic-Kummer extensions K(ζnt,Gn)/K(ζnt) K\left( {{\zeta _{nt}},\root n \of G } \right)/K\left( {{\zeta _{nt}}} \right) for all positive integers n and t, and we describe an explicit finite procedure to compute at once the degree of all these extensions.