Quadratic forms of height and degree 2
type: Item
instance of: scholarly article
title: Quadratic forms of height and degree 2
zbMATH Open document ID: 0866.11030
DOI: 10.1016/0019-3577(96)88656-3
published in: Indagationes Mathematicae. New Series
publication date: 1997-07-20T00:00:00Z
review text: Let \(F\) be a field of characteristic not 2. The height \(h(q)\) of a quadratic form \(q\) over \(F\) is defined by: (i) \(h(q)=h(q_{an})\), where \(q_{an}\) is the anisotropic part of \(q\); (ii) If \(q\) is anisotropic \(h(q)=0\) if and only if \(\dim q\leq 1\); (iii) If \(q\) is anisotropic and of dimension \(\geq 2\), \(h(q)= h(q_{F(q)})+1\) (with \(F(q)\) the function field of the quadratic associated to \(q\)). The degree \(\deg(q)\) satisfies by definition: (i) \(\deg(q)= \deg(q_{an})\); (ii) \(\deg(q)=0\) if \(\dim q\) is odd; (iii) If \(q\) is anisotropic \(\dim q\) is even and if \(h(q)=1\) then \(\dim q=2^\delta\). If this is the case \(\deg q=\delta\); (iv) If \(q\) is anisotropic and \(h(q)>1\), \(\deg(q)= \deg(q_{F(q)})\). The main theorem of the paper classifies all forms of height 2 and degree 2, thereby completing earlier results of Knebusch and Fitzgerald. It is shown that an anisotropic form \(q\) with \(h(q)= \deg(q)=2\) is either excellent or a 6-dimensional Albert form or an 8-dimensional form with nontrivial Clifford invariant, isometric to \(q\otimes\theta\) with \(\theta\) a 1-fold Pfister form. Moreover, it is shown that non-excellent forms of height 2 and degree \(\leq 2\) can be parametrized by cohomological invariants. For forms of degree \(>2\) some partial results are obtained.
reviewed by: Jan Van Geel
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 885621
zbMATH Keywords: splitting tower
zbMATH Keywords: height of a quadratic form
zbMATH Keywords: anisotropic form
zbMATH Keywords: forms of height and degree two
zbMATH Keywords: cohomological invariants
author: Bruno Kahn
MaRDI profile type: Publication
full work available at URL: 0019-3577(96)88656-3
OpenAlex ID: W2016388753
cites work: Cohomologische Invarianten quadratischer Formen
cites work: Q3220649
cites work: The Graded Witt Ring and Galois Cohomology. II
cites work: Über die Grade quadratischer Formen
cites work: Proof of Krull's intersection theorem for the Witt ring
cites work: Pfister forms and \(K\)-theory of fields
cites work: Q4165447
cites work: Quadratic Forms of Height Two
cites work: Q3141464
cites work: Degree four cohomological invariants for quadratic forms
cites work: Generic Splitting of Quadratic Forms, I
cites work: Generic Splitting of Quadratic Forms, II
cites work: Q3877755
cites work: Q3960975
cites work: SIMPLE ALGEBRAS AND QUADRATIC FORMS
cites work: THE NORM RESIDUE HOMOMORPHISM OF DEGREE THREE
cites work: Algebraic \(K\)-theory and quadratic forms. With an appendix by J. Tate
cites work: On quadratic forms isotropic over the function field of a conic
cites work: Q5750935
cites work: Q4713566
description: scientific article; zbMATH DE number 885621