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arXiv · 1805.12588

Tame Galois module structure revisited

Fabio Ferri·Cornelius Greither
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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.

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Fabio Ferri, Cornelius Greither. 2018-05-31. Tame Galois module structure revisited. https://arxiv.org/abs/1805.12588

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