A note on the u-invariant of fields
type: Item
instance of: scholarly article
title: A note on the u-invariant of fields
zbMATH Open document ID: 0563.10018
DOI: 10.1007/BF01237859
author: Enzo R. Gentile
published in: Archiv der Mathematik
publication date: 1985-01-01T00:00:00Z
review text: The u-invariant of a field K of characteristic \(\neq 2\) is defined to be \(\infty\) or the maximum of the dimensions of anisotropic torsion forms over K. The author reproves a well known theorem of \textit{R. Elman} and \textit{T. Y. Lam} [Invent. Math. 21, 125-137 (1973; Zbl 0267.10029)], which states that if quaternion algebras over K form a subgroup in the Brauer group B(K), then the u-invariant of K is equal to 1, 2, 4 or 8. The original proof uses the Arason-Pfister Hauptsatz, which in prevalent opinion is considered to be a strong tool. The main goal of this paper is to give an elementary proof, which does not involve generic methods.
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 12E99
zbMATH DE Number: 3897027
zbMATH Keywords: local-global principle
zbMATH Keywords: linked fields
zbMATH Keywords: linkage
zbMATH Keywords: quadratic form
zbMATH Keywords: Witt ring
zbMATH Keywords: u-invariant
zbMATH Keywords: quaternion algebras
zbMATH Keywords: Brauer group
reviewed by: Andrzej Sładek
MaRDI profile type: Publication
cites work: Quadratic forms and the u-invariant
cites work: Generic Splitting of Quadratic Forms, II
full work available at URL: https://doi.org/10.1007/bf01237859
OpenAlex ID: W2312463666
description: scientific article; zbMATH DE number 3897027