Algebraic systems of quadratic forms of number fields and function fields
type: Item
instance of: scholarly article
title: Algebraic systems of quadratic forms of number fields and function fields
zbMATH Open document ID: 0689.10027
DOI: 10.1007/BF01168301
author: Martin Krüskemper
published in: Manuscripta Mathematica
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155436
review text: If \(F\subset L\) is a finite extension of fields of characteristic \(\neq 2\) then a quadratic form \(\phi\) over \(F\) is algebraic (resp. a scaled trace form) if \(\phi\) is Witt equivalent to the trace form \(\text{Tr}_{L/F}(<1>)\) (resp. to \(\text{Tr}_{L/D}(<\lambda >)\) for some \(\lambda \in L^*)\). Given a form \(\phi\) over \(F\) one wants to decide if \(\phi\) is algebraic. For some classes of fields (e.g. number fields, function fields in one variable over a real closed field) a simple criterion is known. Essentially the question is raised for systems of forms: If \(\phi_1,\dots,\phi_n\) are forms over \(F\), when do there exist a finite field extension \(F\subset L\) and \(\lambda_1=1\), \(\lambda_2,\dots,\lambda_n\in L^*\) such that \(\phi_i\) is Witt equivalent to \(\text{Tr}_{L/F}(<\lambda_i>)?\) An answer is provided for the classes of fields mentioned above.
reviewed by: Niels Schwartz
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 4128901
zbMATH Keywords: algebraic field extension
zbMATH Keywords: Hilbertian field
zbMATH Keywords: number field
zbMATH Keywords: real closed field
zbMATH Keywords: function field
zbMATH Keywords: algebraic quadratic form
zbMATH Keywords: scaled trace form
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4128901