Ultrasoluble coverings of some nilpotent groups by a cyclic group over number fields and related questions

Abstract

Let be a finite nilpotent group of odd order. For every finite cyclic subgroup of odd order we find necessary and sufficient conditions for a class to determine an ultrasoluble extension (under the additional assumption of minimality of all -Sylow subextensions to the extension with class for all non-Abelian -Sylow subgroups of ), that is, for the existence of a Galois extension of number fields with group such that the corresponding embedding problem is ultrasoluble (it has solutions and all such solutions are fields). We also establish a number of related results.