The norm theorem for semisingular quadratic forms
type: Item
instance of: scholarly article
title: The norm theorem for semisingular quadratic forms
zbMATH Open document ID: 1486.11053
author: Ahmed Laghribi
author: Diksha Mukhija
published in: Journal of Pure and Applied Algebra
publication date: 2021-03-03T00:00:00Z
review text: Let \(\varphi\) be a quadratic form over a field \(F\), let \(X=(X_1,\ldots, X_n)\) be an \(n\)-tuple of variables and let \(p=p(X)\) be an irreducible polynomial in \(F[X]\). A classical problem in the algebraic theory of quadratic forms concerns finding necessary and sufficient conditions for \(p\) to be a similarity factor of \(\varphi\) over \(F(X)\), i.e., for \(\varphi\cong p\varphi\) to hold over \(F(X)\), in which case \(p\) is called a norm of \(\varphi\). Without loss of generality, one may assume \(\varphi\) to be anisotropic and \(p\) to have leading coefficient \(1\), say, with respect to the lexicographical ordering of monomials. If the characteristic of \(F\) is not \(2\), the answer was given by \textit{M. Knebusch} [Acta Arith. 24, 279--299 (1973; Zbl 0287.15010)]. He showed that \(p\) is a norm of \(\varphi\) if and only if \(\varphi\) becomes hyperbolic over \(F(p)\), the function field of \(p\), i.e., the field of fractions of the integral domain \(F[X]/(p)\). Actually, Knebusch proved an analogous result for bilinear forms in any characteristic (with hyperbolicity replaced by metabolicity), so it naturally carries over to quadratic forms in characteristic not \(2\) where bilinear and quadratic forms are essentially the same and metabolicity amounts to hyperbolicity. Let us from now on assume that the characteristic of \(F\) is \(2\) in which case the situation becomes more subtle. Here, a quadratic form \(\varphi\) decomposes as an orthogonal sum \(\varphi\cong R\perp\hbox{ql}(\varphi)\) where \(R\) is nonsingular and \(\hbox{ql}(\varphi)\) is the restriction of \(\varphi\) to its radical. In particular, \(R\) is isometric to an orthogonal sum of binary forms of type \([a,b]=ax^2+xy+by^2\), \(a,b\in F\) and thus \(\varphi\) is of dimension \(2r\) for some nonnegative integer \(r\), and \(\hbox{ql}(\varphi)\) is a diagonal form \(\langle a_1,\ldots,a_s\rangle=a_1x_1^2+\ldots+a_sx_s^2\), \(a_i\in F\), and is called the quasilinear part of \(\varph…
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E08
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 7317121
zbMATH Keywords: quadratic form
zbMATH Keywords: quasi-hyperbolicity
zbMATH Keywords: norm theorem
zbMATH Keywords: function field of an irreducible polynomial
zbMATH Keywords: Scharlau transfer
reviewed by: Detlev Hoffmann
MaRDI profile type: Publication
full work available at URL: j.jpaa.2020.106601
OpenAlex ID: W3097670232
cites work: The norm theorem for quadratic forms over a field of characteristic 2
cites work: Q3903156
cites work: Q3516073
cites work: Quadratic forms and Pfister neighbors in characteristic 2
cites work: Q4665009
cites work: Specialization of quadratic and symmetric, bilinear forms, and a norm theorem
cites work: The norm theorem for totally singular quadratic forms
cites work: On the norm theorem for semisingular quadratic forms
cites work: Q3708857
cites work: Q3253825
DOI: 10.1016/J.JPAA.2020.106601
description: scientific article; zbMATH DE number 7317121