Tame Galois module structure revisited

TL;DR

This paper investigates the structure of Galois modules over number fields, focusing on G-Leopoldt fields with weak NIB conditions, providing new limitations and classifications, and correcting previous oversights in the field.

Contribution

It introduces new results on G-Leopoldt fields, especially for specific groups G, and refines the understanding of Hilbert-Speiser fields by correcting prior work.

Findings

Most G-Leopoldt fields are limited or finite in number for certain groups G.

The paper provides exhaustive classifications or finiteness results for some G-Leopoldt fields.

It corrects a previous oversight regarding Hilbert-Speiser fields in recent literature.

Abstract

A number field $K$ is Hilbert-Speiser if all of its tame abelian extensions $L/K$ admit NIB (normal integral basis). It is known that $\mathbb{Q}$ is the only such field, but when we restrict $\text{Gal}(L/K)$ to be a given group $G$, the classification of $G$-Hilbert-Speiser fields is far from complete. In this paper, we present new results on so-called $G$-Leopoldt fields. In their definition, NIB is replaced by ``weak NIB'' (defined below). Most of our results are negative, in the sense that they strongly limit the class of $G$-Leopoldt fields for some particular groups $G$, sometimes even leading to an exhaustive list of such fields or at least to a finiteness result. In particular we are able to correct a small oversight in a recent article by Ichimura concerning Hilbert-Speiser fields.