Steinitz classes of some abelian and nonabelian extensions of even degree

TL;DR

This paper investigates the properties of Steinitz classes in number field extensions with even degree, focusing on their realizability for abelian and nonabelian groups, and reduces the problem to cyclic groups of 2-power degree.

Contribution

It extends previous work to even order groups, showing that studying cyclic groups of 2-power degree suffices for understanding abelian groups' Steinitz classes.

Findings

Realizable Steinitz classes for abelian groups can be studied via cyclic groups of 2-power degree.

The paper develops new results for groups of even order.

It supports the conjecture that the set of realizable classes forms a group.

Abstract

The Steinitz class of a number field extension K/k is an ideal class in the ring of integers O_k of k, which, together with the degree [K:k] of the extension determines the O_k-module structure of O_K. We call R_t(k,G) the classes which are Steinitz classes of a tamely ramified G-extension of k. We will say that those classes are realizable for the group G; it is conjectured that the set of realizable classes is always a group. In this paper we will develop some of the ideas contained in arXiv:0910.5080 to obtain some results in the case of groups of even order. In particular we show that to study the realizable Steinitz classes for abelian groups, it is enough to consider the case of cyclic groups of 2-power degree.