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Summer 2011 On the restricted Hilbert–Speiser and Leopoldt properties
Nigel P. Byott, James E. Carter, Cornelius Greither, Henri Johnston
Illinois J. Math. 55(2): 623-639 (Summer 2011). DOI: 10.1215/ijm/1359762405

Abstract

Let $G$ be a finite abelian group. A number field $K$ is called a Hilbert-Speiser field of type $G$ if, for every tame $G$-Galois extension $L/K$, the ring of integers $\mathcal{O}_L$ is free as an $\mathcal{O}_K[G]$-module. If $\mathcal{O}_L$ is free over the associated order $\mathcal{A}_{L/K}$ for every $G$-Galois extension $L/K$, then $K$ is called a Leopoldt field of type $G$. It is well known (and easy to see) that if $K$ is Leopoldt of type $G$, then $K$ is Hilbert–Speiser of type $G$. We show that the converse does not hold in general, but that a modified version does hold for many number fields $K$ (in particular, for $K/\mathbb{Q}$ Galois) when $G=C_{p}$ has prime order. We give examples with $G=C_5$ to show that even the modified converse is false in general, and that the modified converse can hold when the original does not.

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Nigel P. Byott. James E. Carter. Cornelius Greither. Henri Johnston. "On the restricted Hilbert–Speiser and Leopoldt properties." Illinois J. Math. 55 (2) 623 - 639, Summer 2011. https://doi.org/10.1215/ijm/1359762405

Information

Published: Summer 2011
First available in Project Euclid: 1 February 2013

zbMATH: 1286.11186
MathSciNet: MR3020699
Digital Object Identifier: 10.1215/ijm/1359762405

Subjects:
Primary: 11R29, 11R33

Rights: Copyright © 2011 University of Illinois at Urbana-Champaign

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Vol.55 • No. 2 • Summer 2011
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