Über die zentrale Klassenzahl eines relativ-galoisschen Zahlkörpers

Abstract

For a Galois extension K over a number field k of finite degree, the central class field K̂ of K over k is defined as the maximum unramified abelian extension over K such that the Galois group of K̂ over K is contained in the center of the Galois group of K̂ over k. In this article, a formula for the central class number of K over k, i.e., the extension degree of K̂ over K, is given. Moreover in the case where K k is of prime power degree, a relation between the central class number and the ordinary class number is treated.