Steinitz classes of tamely ramified nonabelian extensions of odd prime power order

TL;DR

This paper investigates the Steinitz classes of tamely ramified nonabelian extensions of odd prime power degree, completing the classification for groups of order p^3 and providing an alternative proof using class field theory.

Contribution

It advances the understanding of Steinitz classes for nonabelian p-groups of order p^3 and offers a new proof approach based on class field theory.

Findings

Complete classification of realizable Steinitz classes for groups of order p^3.

Established that the set of realizable classes forms a group in these cases.

Provided an alternative proof of previous results using class field 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 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 study some l-groups, where l is an odd prime number. In particular, together with [1] we will complete the study of realizable Steinitz classes for groups of order l^3. We will also give an alternative proof of the results of [1], based on class field theory. [1] C. Bruche. Classes de Steinitz d'extensions non abeliennes de degre p^3. Acta Arith., 137(2):177-191, 2009