Translator Disclaimer
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


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.


Download Citation

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.


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

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

Primary: 11R29, 11R33

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


Vol.55 • No. 2 • Summer 2011
Back to Top