Tame Galois module structure revisited

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.