Linkage property of Witt ring and torsion quadratic forms with trivial Witt invariant
type: Item
instance of: scholarly article
title: Linkage property of Witt ring and torsion quadratic forms with trivial Witt invariant
zbMATH Open document ID: 0863.11028
author: Hidemoto Kashima
published in: Tokyo Journal of Mathematics
publication date: 1996-11-25T00:00:00Z
review text: \textit{R. Elman} and \textit{T. Y. Lam} [Invent. Math. 21, 125-137 (1973; Zbl 0267.10029)] established some bounds on the \(u\)-invariant of a field \(F\) depending on whether the powers of the fundamental ideal \(IF\) of the Witt ring \(WF\) are linked or their intersections with the torsion ideal \(W_tF\) of the Witt ring are linked. Thus, if \(I^2F\) is linked, then the \(u\)-invariant assumes only the values 0, 1, 2, 4 or 8, and if \(I^3F\) is torsion free, then \(u\leq 4\). Also, for a formally real field with \(I^nF\cap W_tF\) linked for all \(n\), the values of the \(u\)-invariant are 0, 2, 4, 8, 16 or 18. The present author obtains some further results in this direction. First, instead of the \(u\)-invariant he considers the \(cu\)-invariant, which is defined to be the maximal dimension of anisotropic torsion forms \(\varphi\) with trivial Witt invariant \(c(\varphi)=1\). He proves that, when \(I^3F\) is linked, then \(cu(F)\) assumes only the values 0, 1, 2, 8 or 16, and if additionally \(I^4F\) is torsion free, then \(cu(F)=0,1,2\) or 8. The second result assumes that for \(n\geq 3\) any torsion form in \(I^nF\) is congruent to an \(n\)-fold Pfister form \(\text{mod } I^{n+1}F\). Then the \(cu\)-invariant of a formally real field \(F\) assumes one of the values 0, 2, 8, 16 or 18, and if additionally \(I^4F\) is torsion free, then only 0, 2 or 8 are possible.
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 11E04
zbMATH DE Number: 927037
zbMATH Keywords: linkage property
zbMATH Keywords: torsion quadratic forms
zbMATH Keywords: maximal dimension of anisotropic torsion forms with trivial Witt invariant
zbMATH Keywords: Witt ring
zbMATH Keywords: \(cu\)-invariant
zbMATH Keywords: \(n\)-fold Pfister form
zbMATH Keywords: formally real field
reviewed by: Kazimierz Szymiczek
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270043223
OpenAlex ID: W2066916416
DOI: 10.3836/TJM/1270043223
Recommended article: Q3824477
Recommended article: Q4189339
Recommended article: Q3887533
Recommended article: Q4856900
Recommended article: Q4875290
Recommended article: Quelques remarques sur le $u$-invariant
Recommended article: Some versions of the \(u\)-invariant of a field
Recommended article: Q4515888
Recommended article: The \(u\)-invariant of \(p\)-adic function fields
Recommended article: Q3735780
description: scientific article; zbMATH DE number 927037