Sums of Hermitian squares
type: Item
instance of: scholarly article
title: Sums of Hermitian squares
zbMATH Open document ID: 0644.10017
DOI: 10.1016/0021-8693(88)90273-6
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: The level of a commutative ring R is the least integer n for which \(-1\) is a sum of n squares in R. In this paper the author defines the hermitian level of a ring with involution \(x\to \bar x\) in the same way except that he considers hermitian squares \(x\bar x\) instead of squares. Among others the following results are proved. There exist commutative rings with non-trivial involution having any positive integer as hermitian level. For fields with non-trivial involution or quaternion division algebras with the standard involution, the level, if finite, is a power of two. The case of a quaternion division algebra with a non-standard involution is left open. The paper ends with some remarks about the finiteness of the level for quaternion division algebras over ordered fields.
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 11E08
Mathematics Subject Classification ID: 11E10
Mathematics Subject Classification ID: 13A99
zbMATH DE Number: 4049716
zbMATH Keywords: commutative ring
zbMATH Keywords: hermitian level
zbMATH Keywords: ring with involution
zbMATH Keywords: hermitian squares
zbMATH Keywords: standard involution
zbMATH Keywords: quaternion division algebra
author: D. W. Lewis
reviewed by: M.-A. Knus
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90273-6
OpenAlex ID: W2088318827
cites work: Über die Stufe eines semi-lokalen Ringes
cites work: Quadratic forms over semilocal rings
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description: scientific article; zbMATH DE number 4049716