Representation of polynomials as products of two values of a quadratic form
type: Item
instance of: scholarly article
title: Representation of polynomials as products of two values of a quadratic form
zbMATH Open document ID: 1285.11066
published in: Israel Journal of Mathematics
publication date: 2012-11-19T00:00:00Z
review text: Let \(k\) be a field of characteristic different from \(2\) and \(p\) a monic irreducible polynomial over \(k\). For a quadratic form \(\varphi\) over \(k\), the author considers the following statement (A): if \(\varphi\) is isotropic over the residue field \(k_p = k[t]/p(t)\), then the form \(\varphi \perp -p(t) \varphi\) is isotropic over the rational function field \(k(t)\). While (A) holds true for quadratic polynomials and for odd degree polynomials, the author proves that for a suitably chosen base field \(k\) and any \(n \geq 2\) there exists an irreducible monic polynomial \(p\) of degree \(2n\) for which (A) is false. In addition, the anisotropic part of the form \(\varphi_{k_p}\) is not defined over \(k\), so that \(k_p/k\) is a nonexcellent field extension of degree \(2n\). A weaker variant of (A), to which the author does not know any counterexamples, is also considered. This amounts to requiring that, if \(\varphi\) becomes isotropic over a finite field extension \(\ell\) of \(k\), then there exists a monic irreducible polynomial \(p\) over \(k\) such that \(\ell = k_p\) and the form \(\varphi \perp -p(t) \varphi\) is isotropic over \(k(t)\). It is proved that this holds when \(\ell\) is a biquadratic extension of \(k\) and \(\varphi\) is a form of dimension \(4\).
reviewed by: Kazimierz Szymiczek
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 11E04
zbMATH DE Number: 6108400
zbMATH Keywords: excellent field extension
zbMATH Keywords: biquadratic extension
zbMATH Keywords: quadratic forms over function fields
author: A. S. Sivatski
MaRDI profile type: Publication
full work available at URL: s11856-011-0139-9
OpenAlex ID: W1966592287
cites work: Quadratic forms becoming isotropic on an extension
cites work: Q3160258
cites work: Q3708857
cites work: Q5703059
DOI: 10.1007/S11856-011-0139-9
description: scientific article; zbMATH DE number 6108400