Illinois Journal of Mathematics

On the restricted Hilbert–Speiser and Leopoldt properties

Nigel P. Byott, James E. Carter, Cornelius Greither, and Henri Johnston

Full-text: Open access

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.

Article information

Source
Illinois J. Math., Volume 55, Number 2 (2011), 623-639.

Dates
First available in Project Euclid: 1 February 2013

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1359762405

Digital Object Identifier
doi:10.1215/ijm/1359762405

Mathematical Reviews number (MathSciNet)
MR3020699

Zentralblatt MATH identifier
1286.11186

Subjects
Primary: 11R33: Integral representations related to algebraic numbers; Galois module structure of rings of integers [See also 20C10] 11R29: Class numbers, class groups, discriminants

Citation

Byott, Nigel P.; Carter, James E.; Greither, Cornelius; Johnston, Henri. On the restricted Hilbert–Speiser and Leopoldt properties. Illinois J. Math. 55 (2011), no. 2, 623--639. doi:10.1215/ijm/1359762405. https://projecteuclid.org/euclid.ijm/1359762405


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