Steinitz classes of tamely ramified Galois extensions of algebraic number fields

TL;DR

This paper investigates the Steinitz classes of tamely ramified Galois extensions of number fields, proving a conjecture for a broad class of groups and explicitly determining classes for dihedral groups using class field theory.

Contribution

It proves the realizability conjecture for A'-groups of odd order and determines Steinitz classes for dihedral groups, expanding known cases with a class field theory approach.

Findings

Conjecture holds for A'-groups of odd order.

Explicit description of Steinitz classes for dihedral groups.

Systematic use of class field theory instead of Kummer theory.

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 rt(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. We define A'-groups inductively, starting by abelian groups and then considering semidirect products of A'-groups with abelian groups of relatively prime order and direct products of two A'-groups. Our main result is that the conjecture about realizable Steinitz classes for tame extensions is true for A'-groups of odd order; this covers many cases not previously known. Further we use the same techniques to determine rt(k,D_n) for any odd integer n. In contrast with many other papers on the subject, we systematically use class field theory (instead of Kummer theory and cyclotomic descent).