The rational characterization of certain sets of relatively Abelian extensions

Abstract

Let H be a class group— in the sense of class-field theory— in the rational field P, whose order is some power of a prime l . With H there is associated an Abelian extension K of P. The purpose of this paper is to determine in rational terms and for all fields K given in the described manner, the set T(K/P) of cyclic extensions A of K of relative degree l , which are absolutely normal. In particular we shall find the ramification laws for these fields A, and the possible extension types of a group of order l by the Galois group of K, which are realized in Galois groups of fields in T(K/P). It is fundamental to the programme outlined, that we aim at obtaining purely rational criteria of determination.