Witt rings and almost free pro-2-groups
type: Item
instance of: scholarly article
title: Witt rings and almost free pro-2-groups
zbMATH Open document ID: 0706.11022
DOI: 10.1016/0021-8693(90)90136-C
author: Roger Ware
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: A field F is said to be quasi-pythagorean if the Kaplansky radical of F consists of all nonzero sums of two squares (or, equivalently, if \(I^ 2F\) is torsion free). It is proved here that, if F is a formally real quasi-pythagorean field and L/F is a maximal 2-extension such that \(\sum \dot F^ 2/\dot F^ 2\to\) \(\dot L^ 2/\dot L^ 2\) is injective, then the natural map WF \(\to\) \(W_{red}F\times WL\) is an isomorphism. Here WF, \(I^ 2F\) and \(W_{red}F\) are the Witt ring of F, the square of the fundamental ideal of WF and the reduced Witt ring of F. If moreover, F satisfies a finiteness condition on the space of orderings, then it follows that Gal(F(2)/F) is the free pro-2-product of Gal(K(2)/K) and \(Gal(F_{py}/F)\). Here F(2) is the quadratic closure of F and \(F_{py}\) is the Pythagorean closure of F. These results are used to give a new proof of a structure result for almost free pro-2-groups originally proved by \textit{Yu. L. Ershov} [Mat. Zametki 36, 913-923 (1984; Zbl 0573.12008)].
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 12F10
zbMATH DE Number: 4158719
zbMATH Keywords: formally real quasi-pythagorean field
zbMATH Keywords: Witt ring
zbMATH Keywords: fundamental ideal
zbMATH Keywords: reduced Witt ring
zbMATH Keywords: almost free pro-2-groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4158719