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Annali di Matematica Pura ed Applicata (1923 -)

, Volume 198, Issue 6, pp 2029–2042 | Cite as

Tame Galois module structure revisited

  • Fabio FerriEmail author
  • Cornelius Greither
Article
  • 80 Downloads

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.

Keywords

Class groups Group rings Galois modules Tame abelian extensions 

Mathematics Subject Classification

11R33 11R29 

Notes

Acknowledgements

The majority of the results were obtained in the first author’s M.Sc. thesis, written under the joint supervision of Ilaria Del Corso at Pisa and the second author. Funding was provided by the University of Exeter.

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ExeterExeterUK
  2. 2.Facultät INFUniversität der Bundeswehr MünchenNeubibergGermany

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