On real local-global principles
type: Item
instance of: scholarly article
title: On real local-global principles
zbMATH Open document ID: 0692.12006
DOI: 10.1007/BF02570864
author: Martin Krüskemper
published in: Mathematische Zeitschrift
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/183761
review text: Let \(F\) be a formally real field. For every ordering \(P\) of \(F\) let \(F_ P\) denote a real closure of \(F\) at \(P\). Let \(W_ T\) be the torsion ideal and \(I\) the augmentation ideal of the Witt ring of \(F\). Let \(k_{n_ T} F\) be the subgroup of the Milnor functor \(k_ n F\) generated by all \(\ell (s)\ell (a_ 2)...\ell (a_ n)\) where \(s\) is a sum of squares. \textit{M. Marshall} [Can. J. Math. 29, 606--614 (1977; Zbl 0332.12101)] stated the following two conjectures: (1) The kernel of \(k_ n F\to \prod k_ n F_ P \) is \(k_{n_ T} F\), (2) \(I^ n\cap W_ T=I^{n-1} W_ T\). The main results of this paper are: If the stability index of \(F\) is less than or equal to \(k\) then (1) and (2) hold for \(n\geq k\). If (1) holds for all finitely generated extensions of \(\mathbb Q\) then (1) and (2) hold in general.
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 12J15
zbMATH DE Number: 4132291
zbMATH Keywords: formally real field
zbMATH Keywords: Milnor functor
zbMATH Keywords: stability index
MaRDI profile type: Publication
cites work: Pfister forms and \(K\)-theory of fields
cites work: Q3813862
cites work: Algebraic \(K\)-theory and quadratic forms. With an appendix by J. Tate
OpenAlex ID: W2025512640
description: scientific article; zbMATH DE number 4132291