On fields with finite Brauer groups
type: Item
instance of: scholarly article
title: On fields with finite Brauer groups
zbMATH Open document ID: 0868.12005
author: Ido Efrat
published in: Pacific Journal of Mathematics
publication date: 1997-07-24T00:00:00Z
full work available at URL: v177no1.html
review text: Let \(K\) be a field of characteristic \(\neq 2\), let \(\text{Br} (K)_2\) be the 2-primary part of its Brauer group, and let \(G_K (2)= \text{Gal} (K(2)/ K)\) be the maximal pro-2 Galois group of \(K\). We show that \(\text{Br} (K)_2\) is a finite elementary abelian 2-group \((\mathbb{Z}/ 2\mathbb{Z})^r\), \(r\in \mathbb{N}\), if and only if \(G_K (2)\) is a free pro-2 product of a closed subgroup \(H\) which is generated by involutions and of a free pro-2 group. Thus, the fixed field of \(H\) in \(K(2)\) is pythagorean. The rank \(r\) is in this case determined by the behaviour of the orderings of \(K\). E.g., it is shown that if \(r\leq 6\) then \(K\) has precisely \(r\) orderings, and if \(r< \infty\) then \(K\) has only finitely many orderings.
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 12J15
Mathematics Subject Classification ID: 12F99
zbMATH DE Number: 950189
zbMATH Keywords: Galois cohomology
zbMATH Keywords: ordered fields
zbMATH Keywords: 2-primary part of Brauer group
zbMATH Keywords: maximal pro-2 Galois group
zbMATH Keywords: finite elementary abelian 2-group
zbMATH Keywords: orderings
MaRDI profile type: Publication
OpenAlex ID: W2109105288
DOI: 10.2140/PJM.1997.177.33
description: scientific article; zbMATH DE number 950189