Function fields of Pfister neighbours
type: Item
instance of: scholarly article
title: Function fields of Pfister neighbours
zbMATH Open document ID: 0841.11015
author: Hamza Ahmad
author: Jack Ohm
published in: Journal of Algebra
publication date: 1996-07-16T00:00:00Z
review text: A Pfister neighbor is a quadratic form \(Q\) of dimension \(m\) similar to a subform of a \(2n\)-dimensional Pfister form with \(n<m\). The authors introduce special Pfister neighbors. These are forms \(Q\) similar to forms of the shape \(P_0\perp aP_1\), where \(P_0\) is a Pfister form and \(P_1\) is a nonzero subform of \(P_0\). The Pfister form \(P=P_0\perp aP_0\) is uniquely determined by \(Q\) and is said to be the associated Pfister form of \(Q\) (and \(Q\) is a special Pfister neighbor of \(P\)). It is proved here that for an anisotropic Pfister form \(P\) any two special Pfister neighbors of \(P\) with equal dimensions have the same function field. (It is unknown whether this statement remains true after dropping the assumption that the neighbors are special.) An example is given of an 8-dimensional Pfister form with two 5-dimensional anisotropic special Pfister neighbors which are not similar. This produces an example of 5-dimensional anisotropic forms which have the same function field but are not similar. On the other hand, \textit{A. R. Wadsworth} [Trans. Am. Math. Soc. 208, 352-358 (1975; Zbl 0336.15013)]\ proved that two forms of dimension \(\leq 4\) which have the same function field are similar. Another result of the authors state that all Pfister neighbors of codimension \(\leq 4\) of an anisotropic Pfister form are special. An example is given of a codimension 5 Pfister neighbor (of a 16-dimensional Pfister form) which is not special.
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 840723
zbMATH Keywords: special Pfister neighbors
zbMATH Keywords: function fields of quadratic forms
zbMATH Keywords: Pfister forms
zbMATH Keywords: Zariski cancellation
reviewed by: Kazimierz Szymiczek
MaRDI profile type: Publication
full work available at URL: jabr.1995.1371
OpenAlex ID: W1996965094
DOI: 10.1006/JABR.1995.1371
Recommended article: On quadratic forms of height two and a theorem of Wadsworth
Recommended article: Q3773951
Recommended article: A note on neighboring quadratic forms in characteristic 2
Recommended article: Birational equivalence of n-Pfister neighbors
Recommended article: The Zariski problem for function fields of quadratic forms
Recommended article: Q4376859
Recommended article: Isotropy over function fields of Pfister forms
Recommended article: Isotropy of 8-dimensional quadratic forms over function fields of quadrics
Recommended article: Q4698630
Recommended article: Quadratic Forms of Height Two
description: scientific article; zbMATH DE number 840723