Kummer Theory for Multiquadratic or Quartic Cyclic Number Fields

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