Symbol length and stability index
type: Item
instance of: scholarly article
title: Symbol length and stability index
zbMATH Open document ID: 1260.11025
author: Karim Johannes Becher
author: Paweł Gładki
published in: Journal of Algebra
publication date: 2012-09-06T00:00:00Z
Mathematics Subject Classification ID: 11E10
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 12D15
zbMATH DE Number: 6078362
zbMATH Keywords: quadratic form
zbMATH Keywords: Pythagorean field
zbMATH Keywords: abstract Witt ring
zbMATH Keywords: reduced Witt ring
zbMATH Keywords: stability index
zbMATH Keywords: Milnor \(K\)-theory
zbMATH Keywords: symbol length
reviewed by: Detlev Hoffmann
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2011.12.027
OpenAlex ID: W2072074453
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DOI: 10.1016/J.JALGEBRA.2011.12.027
review text: Let \(F\) be a field of characteristic not \(2\). Let \(n\in{\mathbb N}\) and denote by \(k_n(F)\) the Milnor \(K\)-group \(K_n(F)\) modulo \(2\). \(k_n(F)\) is additively generated by ``symbols'' \(\{ a_1,\ldots,a_n\}\), \(a_i\in F^\times\), and by theorems by Voevodsky resp. Orlov-Vishik-Voevodsky it is known that \(k_n(F)\) is isomorphic via canonical homomorphisms to \(I^nF/I^{n+1}F\) (where \(I^nF\) denotes the \(n\)-th power of the fundamental ideal \(IF\) in the Witt ring \(WF\)) and to \(H^n(F,{\mathbb Z}/2{\mathbb Z})\), the \(n\)-th Galois cohomology group of \(F\) with coefficients in \({\mathbb Z}/2{\mathbb Z}\) (these statements were formerly known as the Milnor Conjectures). The \(n\)-symbol length \(\lambda_n(F)\) is defined to be the smallest nonnegative integer \(m\) such that each element in \(k_n(F)\) can be written as a sum of \(m\) symbols, provided such an integer exists, and \(\lambda_n(F)=\infty\) otherwise. If \(F\) is real and \(T\) is a preordering on \(F\) with \([F^\times:T^\times]=2^{n+1}\), and \(X_T\) is the space of those orderings on \(F\) that contain \(T\), then it is known that \(|X_T|\leq 2^n\), and \(T\) is called a fan of degree \(n\) if equality holds. The (reduced) stability index \(\text{st}(F)\) is defined to be the supremum of the degrees of all the fans of \(F\).NEWLINENEWLINEThe main purpose of the present paper is to relate \(\text{st}(F)\) to \(\lambda_n(F)\) for a real field \(F\), and more generally in the context of an abstract Witt ring with suitable adaptations of the definitions of the symbol length and the stability index. The results for fields read as follows. If \(\lambda_i(F)<\infty\) for some \(i\geq 2\) then \(\text{st}(F)<\infty\), in particular \(\text{st}(F)\leq 2\lambda_2(F)-1\). If \(F\) is Pythagorean, then one also has that \(s=\text{st}(F)<\infty\) implies \(\lambda_2(F)<\infty\). More precisely, one has \(\lambda_2(F)=s\) if \(1\leq s\leq 2\), and \([s/2]+1\leq \lambda_2(F)\leq 2^{s-1}(2^{s-2}-1)…
description: scientific article; zbMATH DE number 6078362