The square class invariant
type: Item
instance of: scholarly article
title: The square class invariant
zbMATH Open document ID: 0707.11024
author: Craig M. Cordes
author: David L. Foreman
published in: Rocky Mountain Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: For a field F and a quadratic form Q over F, the square class invariant for Q is the function \(f_ Q: \dot F/\dot F^ 2\to {\mathbb{Z}}\) defined by \(f_ Q(a\dot F^ 2)=n\) if \(Q\cong n+P\) and a is not represented by P. If F is non-formally real the level of F, s(F), is the smallest number of squares adding to -1. The following are proved: If F is non- pythagorean and s(F)\(\geq 4\) then \(f_ Q\) does not classify forms over F. F is a C-field if its anisotropic forms are uniquely determined by their value sets. If s(F)\(\leq 2\) and \(f_ Q\) classifies forms over F, then F is a C-field. If F is a C-field with \(s(F)=1\) or 2 and either u(F) or \(| \dot F/D(<1,1>)|\) is finite, then \(f_ Q\) classifies forms over F. If F is nonreal, there are more than two quaternion algebras over F, and \(u(F)=4\), then \(f_ Q\) and det do not classify forms over F.
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 4160833
zbMATH Keywords: quadratic form
zbMATH Keywords: square class invariant
zbMATH Keywords: C-field
reviewed by: A.F.T.W.Rosenberg
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181073161
OpenAlex ID: W2055442749
cites work: Quadratic forms over fields with four quaternion algebras
cites work: The Witt group and the equivalence of fields with respect to quadratic forms
cites work: Fröhlich's local quadratic forms.
cites work: Q5675788
cites work: The square class invariant for Pythagorean fields
cites work: The square class invariant for quadratic forms and the classification problem
DOI: 10.1216/RMJM/1181073161
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description: scientific article; zbMATH DE number 4160833