A question on the discriminants of involutions of central division algebras
type: Item
instance of: scholarly article
title: A question on the discriminants of involutions of central division algebras
zbMATH Open document ID: 0803.16013
DOI: 10.1007/BF01459519
author: R. Sridharan
author: R. Parimala
author: Venapally Suresh
published in: Mathematische Annalen
publication date: 1995-01-03T00:00:00Z
full work available at URL: https://eudml.org/doc/165148
review text: Let \(D\) be a finite-dimensional central simple algebra over a field \(k\) of characteristic not 2 with a \(k\)-linear involution \(\sigma\) of orthogonal type. As for quadratic forms, one can associate to \(\sigma\) a discriminant with values in \(k^ \times/{k^ \times}^ 2\). Assume that \(\dim D \geq 16\). If \(D\) is not a division algebra, it is easy to see that the set of discriminants of involutions of orthogonal type is the full group of reduced norms from \(D^ \times\) modulo squares in \(k^ \times\). In this beautiful short note, the authors show that this is also the case if \(D\) is a division algebra. Consequences are the (surprising) facts that the set of discriminants of involutions of orthogonal type is always a group and that \(D\) admits an involution of discriminant \(1\) (if it admits any). In fact the main result is that, for any \(D\) of dimension \(\geq 16\), whose class in the Brauer group is 2-torsion, every element in \(D\) is symmetric for an involution of orthogonal type whose discriminant is \(1\). The proof is in two steps: the case \(\dim D = 16\) and the reduction to this case. The case \(\dim D = 16\) was already known [see \textit{M.-A. Knus, T. Y. Lam, D. B. Shapiro} and \textit{J.-P. Tignol}: Discriminants of involutions on biquaternion algebras, in Am. Math. Soc. Proc. Santa Barbara (1992)], but the authors also give a very simple proof in this case.
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 12E15
Mathematics Subject Classification ID: 11E04
zbMATH DE Number: 537333
zbMATH Keywords: finite-dimensional central simple algebra
zbMATH Keywords: quadratic forms
zbMATH Keywords: discriminant
zbMATH Keywords: involutions of orthogonal type
zbMATH Keywords: reduced norms
zbMATH Keywords: Brauer group
reviewed by: M.-A. Knus
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 537333