Characterization of Hermitian trace forms
type: Item
instance of: scholarly article
title: Characterization of Hermitian trace forms
zbMATH Open document ID: 0930.11023
author: Grégory Berhuy
published in: Journal of Algebra
publication date: 1999-04-07T00:00:00Z
review text: Let \(q\) be a nondegenerate quadratic form in the even number \(m\) of variables over the field \(K\) of characteristic different from 2. \textit{M. Epkenhans} [Arch. Math. 60, 527-529 (1993; Zbl 0777.11009)] has given necessary and sufficient conditions for \(q\) to be of the form \(q(x) = \text{Tr}_{E/K}(x^2)\) for a separable extension \(E\) of \(K\) of degree \(m\). The author modifies this question and asks when is \(q\) of the form \(q(x) = \text{Tr}_{E/K}(x x^{\sigma})\) for an involution \(\sigma\) of \(E/K\) as above. He shows that this is the case if and only if the total signature of \(q\) is nonnegative and either \(m \geq 4\) or \(m = 2\) and \(q \cong \langle 2,-2D\rangle\) holds with \(D \in K^{\times}- (K^{\times})^2\).
Mathematics Subject Classification ID: 11E39
Mathematics Subject Classification ID: 12F10
zbMATH DE Number: 1251416
zbMATH Keywords: trace forms
zbMATH Keywords: hermitian forms
zbMATH Keywords: involution
reviewed by: Rainer Schulze-Pillot
MaRDI profile type: Publication
full work available at URL: jabr.1998.7600
OpenAlex ID: W2066559134
cites work: Q3344268
cites work: On trace forms of algebraic number fields
cites work: Algebraic number field extensions with prescribed trace form
cites work: Q3708857
cites work: The Witt invariant of the form \(\text{Tr}(x^ 2)\)
DOI: 10.1006/JABR.1998.7600
description: scientific article; zbMATH DE number 1251416