Nonexcellence of multiquadratic field extensions
type: Item
instance of: scholarly article
title: Nonexcellence of multiquadratic field extensions
zbMATH Open document ID: 1133.11027
published in: Journal of Algebra
publication date: 2004-10-01T00:00:00Z
review text: A field extension \(L/F\) is said to be excellent if for any quadratic form \(f\) over \(F\) the anisotropic part of \(f\) over \(L\) is defined over \(F\). Quadratic extensions of \(F\) are known to be excellent and if all multiquadratic extensions of \(F\) are excellent, then Merkurjev's theorem on the injectivity of Clifford invariant follows in an elementary way (that is, with no use of algebraic geometry or \(K\)-theory). For this see \textit{R. Elman} et al. [Proc. Conf. Quadratic Forms, Kingston 1976, Queen's Pap. Pure Appl. Math. 46, 445--492 (1977; Zbl 0385.12010)]. However, it was soon discovered that biquadratic extensions can be nonexcellent, the first counterexamples constructed by \textit{R. Elman} et al. [Am. J. Math. 105, 1119--1170 (1983; Zbl 0492.10014)]. The present author proves that, for any field \(k_0\) of characteristic not \(2\), if \(k_0\) has \(n+1\) multiplicatively independent square classes \(a,b_1,\ldots,b_n\), then there exists a field extension \(F/k_0\) with the property that the multiquadratic extension \(K=F(\sqrt{b_1}, \ldots, \sqrt{b_n})\) of \(F\) is not excellent. A 4-dimensional form \(f\) over \(F\) of determinant \(a\) is constructed, which is isotropic over \(K\) and anisotropic over any subfield of \(K\), and the anisotropic part of \(f\) over \(K\) is not defined over \(F\).
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 12F05
zbMATH DE Number: 2105401
zbMATH Keywords: Quadratic forms
zbMATH Keywords: conic
zbMATH Keywords: \(K\)-groups
author: A. S. Sivatski
reviewed by: Kazimierz Szymiczek
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.07.014
OpenAlex ID: W1972057741
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DOI: 10.1016/J.JALGEBRA.2003.07.014
description: scientific article; zbMATH DE number 2105401