Quadratic forms and Galois cohomology over formally real fields
type: Item
instance of: scholarly article
title: Quadratic forms and Galois cohomology over formally real fields
zbMATH Open document ID: 0777.11012
author: Roger Ware
published in: Journal of Algebra
publication date: 1993-12-12T00:00:00Z
review text: Let \(F\) be a non-Pythagorean field of characteristic different from two and let the third power \(I^ 3 F\) of the fundamental ideal \(IF\) of the Witt ring of \(F\) be torsion free. The main result of the paper is as follows. If \(I^ 3 F(\sqrt{-1})=0\), or, if every binary torsion form represents a totally positive element of the field \(F\), then there exists an exact sequence \(1\to{\mathcal F}\to G_ F\pi\to\mathbb{Z}_ 2\to 1\) where \({\mathcal F}\) is a free pro-2-group and \(G_ F \pi\) is the Galois group of the Pythagorean closure of \(F\) over \(F\). This generalizes earlier results of the author [J. Pure Appl. Algebra 30, 95-107 (1983; Zbl 0518.10024)] valid only for algebraic number fields. As corollaries the author obtains complete characterizations in terms of Galois groups of fields \(F\) with \(I^ 3 F=0\), or \(I^ 2 F\) torsion free, or, under additional hypotheses, of fields with \(I^ 3 F\) torsion free.
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 12G05
zbMATH DE Number: 218571
zbMATH Keywords: formally real fields
zbMATH Keywords: Galois groups of infinite field extensions
zbMATH Keywords: non- Pythagorean field
zbMATH Keywords: fundamental ideal
zbMATH Keywords: Witt ring
zbMATH Keywords: exact sequence
zbMATH Keywords: free pro-2-group
reviewed by: Kazimierz Szymiczek
MaRDI profile type: Publication
full work available at URL: jabr.1993.1125
OpenAlex ID: W1984912057
DOI: 10.1006/JABR.1993.1125
description: scientific article; zbMATH DE number 218571