Infrastructure of ambiguous ideal classes of orders of real quadratic fields
type: Item
instance of: scholarly article
title: Infrastructure of ambiguous ideal classes of orders of real quadratic fields
zbMATH Open document ID: 0756.11030
author: Franz Halter-Koch
author: Pierre Kaplan
author: Kenneth S. Williams
author: Yoshihiko Yamamoto
published in: L'Enseignement Mathématique. 2e Série
publication date: 1992-09-26T00:00:00Z
review text: The authors intend to study ``Shanks infrastructure'' in the orders of a real quadratic field [cf. \textit{D. Shanks}, Proc. 1972 Number Theory Conf. Boulder, Colorado, 217-224 (1972; Zbl 0334.12005)]. Let \({\mathcal O}_{\mathcal D}\) be an order of a real quadratic field with discriminant \({\mathcal D}\). Then, they first define the ``symmetric ideal'', which is necessarily reduced, associated with certain decomposition of \({\mathcal D}\) into the sum of two squares, and prove the following: Let \({\mathcal C}\) be a primitive ambiguous non-principal class of order \({\mathcal O}_{\mathcal D}\) which contains exactly \(\ell\) primitive reduced ideals. Then, in the case of \(N(\varepsilon_{\mathcal D})=-1\), \(\ell\) is odd and \({\mathcal C}\) contains a reduced ambiguous ideal and a symmetric ideal. In the case of \(N(\varepsilon_{\mathcal D})=+1\), \(\ell\) is even and \({\mathcal C}\) contains either two reduced ambiguous ideals or two symmetric ideals. Moreover, they show that the symmetric ideal \({\mathcal S}'\) constructed with the symmetric ideal \({\mathcal S}\) in an ambiguous class in certain simple manner is always principal or equivalent to \({\mathcal S}\) according to \(N(\varepsilon_{\mathcal D})=-1\) or \(+1\).
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R54
zbMATH DE Number: 55794
zbMATH Keywords: ideal classes
zbMATH Keywords: infrastructure
zbMATH Keywords: orders
zbMATH Keywords: real quadratic field
reviewed by: Hideo Yokoi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 55794