Isotropy of quadratic forms over the function field of a quadric
type: Item
instance of: scholarly article
title: Isotropy of quadratic forms over the function field of a quadric
zbMATH Open document ID: 0840.11017
DOI: 10.1007/BF02572626
published in: Mathematische Zeitschrift
publication date: 1996-07-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174821
review text: Let \(F\) be a field of characteristic different from 2. This paper deals with the question whether an anistropic quadratic form \(\varphi\) over \(F\) becomes isotropic over the function field \(F(\psi)\) of another quadratic form \(\psi\) over \(F\). The first main theorem says that if there exists \(n\geq 0\) such that \(\dim\varphi\leq 2^n< \dim\psi\) then \(\varphi\) stays anisotropic over \(F(\psi)\). As an immediate consequence one has that if \(\psi\) is anisotropic with \(\dim\psi= 2^n+ m\), where \(n\geq 0\) and \(1\leq m\leq 2^n\) then the Witt index of \(\psi\otimes F(\psi)\) is less than or equal to \(m\). The second main theorem states that if \(\varphi\) is an anisotropic form over \(F\) with \(\dim \varphi\leq 2^n+ 1\) then there exists a field extension \(K/F\) with an anisotropic \((n+ 1)\)-fold Pfister form \(\pi\) over \(K\) such that \(\varphi\otimes K\) is a subform of \(\pi\). In fact, if there exists an anisotropic \((n+ 1)\)-fold Pfister form \(\pi\) already over \(F\) then one can construct \(K/F\) such that \(\pi\otimes K\) stays anisotropic and \(\varphi\otimes K\) is a subform of \(\pi\otimes K\). An example shows that there exists a field \(F\) with an anisotropic form \(\varphi\) with \(\dim\varphi= 2^n+ 2\), \(n\geq 2\), such that there does not exist \(K/F\) with \(\varphi\otimes K\) an anisotropic Pfister neighbor. This shows that generally the above result cannot be improved. The third main theorem states that if \(n\geq 3\) and \(\psi\) is an anisotropic form over \(F\) with \(\dim\psi= 2^n+ 1\), and if \(\gamma\) is a form over \(F\) with \(\dim\gamma\geq 2^{n+ 1}- 3\) then \(\psi\otimes F(\gamma)\) is isotropic if and only if there exists an \((n+ 1)\)-fold Pfister form \(\rho\) over \(F\) such that \(\psi\) and \(\gamma\) are Pfister neighbors of \(\rho\). The crucial ingredient used in the proofs of these results is a lemma which says that if \(\varphi\) is anisotropic over \(F\) with \(\dim\varphi\leq 2^n\) then there exists …
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 856576
zbMATH Keywords: function field of a quadratic form
zbMATH Keywords: anistropic quadratic form
zbMATH Keywords: Witt index
zbMATH Keywords: Pfister form
zbMATH Keywords: isotropic
zbMATH Keywords: Pfister neighbors
zbMATH Keywords: function field
author: Detlev Hoffmann
reviewed by: Detlev Hoffmann
MaRDI profile type: Publication
cites work: Generic Splitting of Quadratic Forms, I
cites work: Generic Splitting of Quadratic Forms, II
cites work: Q5675788
cites work: SIMPLE ALGEBRAS AND QUADRATIC FORMS
cites work: Generic element in \(SK_ 1\) for simple algebras
cites work: Q3708857
description: scientific article; zbMATH DE number 856576