W-groups and values of binary forms
type: Item
instance of: scholarly article
title: W-groups and values of binary forms
zbMATH Open document ID: 0778.11022
DOI: 10.1016/0022-4049(93)90069-6
author: Tara L. Smith
author: Ján Mináč
published in: Journal of Pure and Applied Algebra
publication date: 1994-01-06T00:00:00Z
review text: The \(W\)-group \(G_ F\) of a field \(F\) is the Galois group over \(F\) of the field \(F^{(3)}\) defined to be the compositum of \(F^{(2)}\) and of all quadratic extensions of \(F^{(2)}\) which are Galois extensions of \(F\), where \(F^{(2)}\) is the compositum of \(F\) and of all quadratic extensions of \(F\). The Witt ring of a field \(F\) determines the \(W\)-group up to isomorphism, and the \(W\)-group determines the Witt ring, unless every element of the field is a sum of two squares. The authors show that, more specifically, the value sets of binary quadratic forms over a field \(F\) can be described explicitly in terms of the maximal abelian quotient of the \(W\)-group. They also give criteria in terms of the \(W\)-group for a field \(F\) to have the level \(s(F)\) not greater than 4.
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 219102
zbMATH Keywords: Galois groups of infinite field extensions
zbMATH Keywords: Witt ring
zbMATH Keywords: value sets of binary quadratic forms
zbMATH Keywords: maximal abelian quotient of the \(W\)-group
zbMATH Keywords: level
reviewed by: Kazimierz Szymiczek
MaRDI profile type: Publication
full work available at URL: 0022-4049(93)90069-6
OpenAlex ID: W2060453735
cites work: Orthogonal representations of Galois groups, Stiefel-Whitney classes and Hasse-Witt invariants.
cites work: Fields with prescribed quadratic form schemes
cites work: Classification of Demushkin Groups
cites work: Q3877755
cites work: Q3897105
cites work: Decomposition of Witt Rings and Galois Groups
cites work: Formally real fields, pythagorean fields, C-fields and W-groups
cites work: Witt rings and Galois groups
cites work: Q4258075
cites work: Galois Cohomology
cites work: Q3856855
description: scientific article; zbMATH DE number 219102