On the Ultrasolvability of p-Extensions of an Abelian Group by a Cyclic Kernel

Abstract

The paper contains a solution of A. V. Yakovlev’s problem in the embedding theory for p-extensions of odd order with a cyclic normal subgroup and an Abelian quotient group: for such nonsplit extensions there exists a realization for the quotient group as a Galois group over number fields such that the corresponding embedding problem is ultrasolvable (i.e., this embedding problem is solvable and has only fields as solutions). A solution for embedding problems of p-extensions of odd order with kernel of order p and with a quotient group that is represented by a direct product of its proper subgroups is also given – this is a generalization for p > 2 of an analogous result for p = 2 due to A. Ledet.