Swan modules and realisable classes for Kummer extensions of prime degree
type: Item
instance of: scholarly article
title: Swan modules and realisable classes for Kummer extensions of prime degree
zbMATH Open document ID: 0923.11152
author: Daniel R. Replogle
published in: Journal of Algebra
publication date: 1999-08-15T00:00:00Z
review text: If one fixes a cyclic group of prime order \(G\) and a number field \(K\), then the set of classes \(R\) of \({\mathbb Z} G\)-modules realisable as \(O_L\) with \(L\) a tame \(G\)-extension of \(K\) forms a subgroup \(R\) of the class group of \({\mathbb Z} G\). McCulloh gave a very explicit description of \(R\) in terms of a Stickelberger ideal. In the present paper, the author proves a result bounding \(R\) from below, based on McCulloh's description. More precisely he shows that if \(K\) contains \(\zeta_l\), then \(R\) contains \(T^{(l-1)/2}\), where \(T\) is the so-called Swan subgroup. The second main result of the paper (which, incidentally, does not use the first one) gives criteria just when \(O_L\) is in \(T\), following an earlier result by Gómez Ayala. Finally, as a consequence of the first theorem, the author deduces that \(R\) is nontrivial if \(K=\mathbb{Q}(\zeta_l)\) with \(l>3\) any prime. A more general result in this direction is proved in a forthcoming joint paper of the author, \textit{K. Rubin, A. Srivastav}, and the reviewer [J. Number Theory (to appear)] for every number field \(K\) except \({\mathbb Q}\), one may find a prime \(l\) (infinitely many in fact) such that \(R\) is nontrivial.
reviewed by: Cornelius Greither
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 12F10
zbMATH DE Number: 1271818
zbMATH Keywords: Kummer extensions
zbMATH Keywords: Swan modules
zbMATH Keywords: integral normal bases
MaRDI profile type: Publication
full work available at URL: jabr.1998.7652
OpenAlex ID: W1970772895
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DOI: 10.1006/JABR.1998.7652
description: scientific article; zbMATH DE number 1271818