On the projective Schur group of a field
type: Item
instance of: scholarly article
title: On the projective Schur group of a field
zbMATH Open document ID: 0852.16011
author: Eli Aljadeff
author: Jack Sonn
published in: Journal of Algebra
publication date: 1996-11-06T00:00:00Z
review text: The authors disprove a conjecture of \textit{P. Nelis} and \textit{F. Van Oystaeyen} [J. Algebra 137, 501-518 (1991; Zbl 0735.13003)] by which the projective Schur group \(PS(k)\) of a field \(k\), as has been defined by \textit{F. Lorenz} and \textit{H. Opolka} [Math. Z. 162, 175-182 (1978; Zbl 0384.16009)], should coincide with the whole Brauer group \(Br(k)\) of \(k\). In the counter-example \(k\) is a rational function field in one variable over an infinite field that is finitely generated over its prime field. Over such fields there exist cyclic algebras of which no splitting field is contained in the composite of a cyclotomic extension and a Kummer extension of \(k\); a projective Schur algebra, however, is shown to admit an abelian splitting field that is a subfield of a radical extension of \(k\).
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 14F22
zbMATH DE Number: 840716
zbMATH Keywords: projective Schur groups
zbMATH Keywords: Brauer groups
zbMATH Keywords: rational function fields in one variable
zbMATH Keywords: cyclic algebras
zbMATH Keywords: Kummer extensions
zbMATH Keywords: projective Schur algebras
zbMATH Keywords: Abelian splitting fields
reviewed by: Jürgen Ritter
MaRDI profile type: Publication
full work available at URL: jabr.1995.1364
OpenAlex ID: W2078107344
DOI: 10.1006/JABR.1995.1364
description: scientific article; zbMATH DE number 840716