On the generic splitting of quadratic forms in characteristic 2
type: Item
instance of: scholarly article
title: On the generic splitting of quadratic forms in characteristic 2
zbMATH Open document ID: 1007.11017
author: Ahmed Laghribi
published in: Mathematische Zeitschrift
publication date: 2002-12-01T00:00:00Z
review text: The theory of generic splitting of quadratic forms goes essentially back to a series of papers by Knebusch in the early to mid 1970s and is at its origin concerned with quadratic forms over fields of characteristic \(\neq 2\) and how they behave after passing from one field to another via places. Of particular interest is the case where the fields involved are extensions of the base field \(F\) over which the form is defined and where the places are \(F\)-places, and function fields of quadratic forms play a central role in this theory. Let \(q\) be a (nonsingular) form over \(F\). The function field \(F(q)\) of \(q\) is the function field of the projective quadric defined by \(q=0\). The isotropy behaviour of \(q\) over field extensions is largely encoded in what is called a generic splitting tower of \(q\) over \(F\), which is defined inductively by \(F_0=F\), \(q_0\) the anisotropic part of \(q\), and for \(i\geq 1\), \(F_i=F_{i-1}(q_{i-1})\) and \(q_i\) the anisotropic part of \(q\) over \(F_i\). By construction, \(\dim q_i<\dim q_{i-1}\), and the smallest \(h\) such that \(\dim q_h\leq 1\) is called the height of \(q\). It turns out that if \(K\) is a field extension of \(F\), then there exists \(i\in\{ 0,\cdots , h\}\) with \(\dim q_i\) equal to the dimension of the anisotropic part of \(q\) over \(K\). Much of this theory goes through also in characteristic \(2\) when considering regular forms (nonsingular forms or forms with nonzero radical of dimension \(1\)) as was shown by \textit{M. Knebusch} and \textit{U. Rehmann} [Contemp. Math. 272, 173-199 (2000; Zbl 0995.11028)]. For arbitrary singular forms, the situation is more complicated and is under current investigation (book in preparation by Knebusch). In the present paper, the author studies the behaviour of (possibly singular) quadratic forms in characteristic \(2\) over the fields in a splitting tower defined as above (which he calls standard splitting tower). If \(q\) is a quadratic form over a field \…
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E88
zbMATH DE Number: 1837067
zbMATH Keywords: quadratic form
zbMATH Keywords: singular quadratic form
zbMATH Keywords: characteristic \(2\)
zbMATH Keywords: function field of a quadric
zbMATH Keywords: generic splitting tower
zbMATH Keywords: excellent quadratic form
zbMATH Keywords: quadratic form of height \(2\)
reviewed by: Detlev Hoffmann
MaRDI profile type: Publication
full work available at URL: s002090200387
OpenAlex ID: W1990134970
DOI: 10.1007/S002090200387
description: scientific article; zbMATH DE number 1837067