On the cohomological characterization of real free pro-\(2\)-groups
type: Item
instance of: scholarly article
title: On the cohomological characterization of real free pro-\(2\)-groups
zbMATH Open document ID: 0849.12012
DOI: 10.1007/BF02567821
author: Antonio Engler
published in: Manuscripta Mathematica
publication date: 1996-11-17T00:00:00Z
full work available at URL: https://eudml.org/doc/156126
review text: A real free pro-2-group is a free pro-2-product of copies of \(\mathbb{Z}/2 \mathbb{Z}\) with a free pro-2-group. The main result of this paper is the following cohomological characterization: a pro-2-group \(G\) is real free if and only if \(G\) admits a free closed subgroup \(H_0\) of index 2 such that the corresponding quadratic character \(\chi_0\) satisfies \(\chi_0 \cup \chi = \chi \cup \chi\) for every \(\chi \in H^1 (G, \mathbb{Z}/2 \mathbb{Z})\). In particular, for any field \(K\) of characteristic \(\neq 2\), the Galois group \(G_K(2)\) of the maximal pro-2-extension of \(K\) is real free if and only if its closed subgroup \(H_0\) corresponding to \(K (\sqrt {-1})\) is free (the additional condition on the character \(\chi_0\) is automatically verified because of classical properties of quaternion algebras). This gives an ample provision of examples of Galois groups which are real free pro-2-groups. Recall that the Hasse invariant \(\widetilde u(K)\) is the maximal dimension of the anisotropic quadratic forms over \(K\) which become isotropic over all (if any) real closures of \(K\). As an application of his criterion, the author proves that for a formally real field \(K\), the following conditions are equivalent: i) \(G_K (2)\) is real free; ii) \(\widetilde u(K) \leq 2\); iii) \(I^2 K = 2IK\) and every sum of squares is a sum of two squares. Here \(IK\) is the usual fundamental ideal of the Witt ring of \(K\).
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 866166
zbMATH Keywords: real free pro-2-group
zbMATH Keywords: cohomological characterization
zbMATH Keywords: Galois groups
zbMATH Keywords: Hasse invariant
zbMATH Keywords: formally real field
zbMATH Keywords: fundamental ideal of the Witt ring
reviewed by: Thong Nguyen Quang Do
MaRDI profile type: Publication
cites work: Cohomologische Invarianten quadratischer Formen
cites work: Q4748231
cites work: Quadratic Forms Over Formally Real Fields and Pythagorean Fields
cites work: On some Hasse principles over formally real fields
cites work: Them-ordered real free pro-2-group. Cohomological characterizations∗
cites work: Galois groups of intersections of local fields
cites work: Cohomology theory of Artin-Schreier structures
cites work: On the cohomological dimension of Artin-Schreier structures
cites work: Q3726225
cites work: Real free groups and the absolute Galois group of \(R(t)\)
cites work: The absolute Galois group of a pseudo p-adically closed field.
cites work: Q5675788
cites work: Remarks on the Pythagoras and Hasse number of real fields.
cites work: Quadratic forms and Galois cohomology
cites work: Profinite Groups, Arithmetic, and Geometry. (AM-67)
cites work: Sur la dimension cohomologique des groupes profinis
cites work: Quadratic forms and profinite 2-groups
OpenAlex ID: W2012267326
description: scientific article; zbMATH DE number 866166