Total positivity and algebraic Witt classes
type: Item
instance of: scholarly article
title: Total positivity and algebraic Witt classes
zbMATH Open document ID: 0589.10021
DOI: 10.1007/BF02567415
author: Robert Perlis
author: Dennis R. Estes
author: Jurgen Hurrelbrink
published in: Commentarii Mathematici Helvetici
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140014
review text: Let E be a finite extension of the algebraic number field F. \(<E>\) denotes the class in the Witt ring W(F) of F given by the trace form \(tr_{E/F}(x^ 2)\). The Witt classes in W(F) arising in this way from algebraic extensions E/F are called algebraic classes. The main result of the paper under review is the following theorem: The element a in \(F^{\#}\) is totally positive in F (i.e. a is positive in every possible ordering of F) if and only if the Witt class of the quadratic form \(a\cdot X^ 2\) in W(F) is algebraic. The proof of the above mentioned theorem uses results of Conner and Perlis concerning trace forms of algebraic number fields and a characterization of totally positive elements in \(F^{\#}\) proved in the first section of the article.
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 11R80
Mathematics Subject Classification ID: 11R70
zbMATH DE Number: 3946223
zbMATH Keywords: Brauer groups
zbMATH Keywords: Milnor K-group
zbMATH Keywords: rank one quadratic form
zbMATH Keywords: norm
zbMATH Keywords: theorem
zbMATH Keywords: finite extension
zbMATH Keywords: Witt ring
zbMATH Keywords: trace forms
zbMATH Keywords: totally positive elements
reviewed by: Hans-Jochen Bartels
MaRDI profile type: Publication
OpenAlex ID: W2084909309
description: scientific article; zbMATH DE number 3946223