Witt rings and Galois groups
type: Item
instance of: scholarly article
title: Witt rings and Galois groups
zbMATH Open document ID: 0861.11030
DOI: 10.2307/2118582
author: Michel Spira
author: Ján Mináč
published in: Annals of Mathematics. Second Series
publication date: 1997-05-21T00:00:00Z
review text: This paper is concerned with the connections between the Witt ring of a field and the structure of certain Galois extensions of that field. In particular, it is shown that the Witt ring determines, and is determined by, the Galois group of a certain 2-extension of the field (with an unavoidable uncertainty over the characteristic of the Witt ring in the case when the form \(\langle 1,1 \rangle\) is universal). This group is called the \(W\)-group of the field \(F\), denoted \({\mathcal G}_F\). It can be viewed as the Galois group over \(F\) of the compositum over \(F\) of all quadratic, cyclic of order 4, and dihedral of order 8 extensions. \(W\)-groups are calculated for several specific fields and classes of fields, and all possible abelian \(W\)-groups are determined. An alternative way of computing \(W\)-groups, using cohomological methods, is given. Finally, the work on \(W\)-groups is put in the context of Merkurjev's result on the isomorphism from \(k_2F\) to \(Br_2F\).
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 12G05
zbMATH DE Number: 935300
zbMATH Keywords: \(W\)-groups
zbMATH Keywords: cohomological methods
zbMATH Keywords: Brauer group
zbMATH Keywords: Witt ring
zbMATH Keywords: Galois extensions
zbMATH Keywords: Galois group
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2118582
OpenAlex ID: W2021226896
description: scientific article; zbMATH DE number 935300