An explicit candidate for the set of Steinitz classes of tame Galois extensions with fixed Galois group of odd order

TL;DR

This paper investigates the set of Steinitz classes of tame Galois extensions with odd order G, proposing an explicit candidate subgroup and proving its equality with the set in specific cases using cohomological techniques.

Contribution

It introduces a subgroup W(k,G) as a candidate for the Steinitz classes set and proves its equality with R_t(k,G) for groups of order dividing l^4, refining previous methods.

Findings

R_t(k,G) is contained in W(k,G) for odd order G.

Equality R_t(k,G)=W(k,G) holds for certain groups of order dividing l^4.

Constructs tame Galois extensions with prescribed Steinitz classes using cohomological methods.

Abstract

Given a finite group G and a number field k, a well-known conjecture asserts that the set R_t(k,G) of Steinitz classes of tame G-Galois extensions of k is a subgroup of the ideal class group of k. In this paper we investigate an explicit candidate for R_t(k,G), when G is of odd order. More precisely, we define a subgroup W(k,G) of the class group of k and we prove that R_t(k,G) is contained in W(k,G). We show that equality holds for all groups of odd order for which a description of R_t(k,G) is known so far. Furthermore, by refining techniques introduced in arXiv:0910.5080v1, we use the Shafarevich-Weil Theorem in cohomological class field theory, to construct some tame Galois extensions with given Steinitz class. In particular, this allows us to prove the equality R_t(k,G)=W(k,G) when G is a group of order dividing l^4, where l is an odd prime.