On trace forms of étale algebras and field extensions
type: Item
instance of: scholarly article
title: On trace forms of étale algebras and field extensions
zbMATH Open document ID: 0821.11024
DOI: 10.1007/BF02571952
author: Martin Krüskemper
author: Martin Epkenhans
published in: Mathematische Zeitschrift
publication date: 1995-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/174698
review text: The investigation of quadratic forms represented as trace forms of finite extension fields was initiated by \textit{P. E. Conner} and \textit{R. Perlis} [A survey of trace forms of algebraic number fields. World Sci., Singapore (1984; Zbl 0551.10017)]. The emphasis was on the characterization of forms which are similar (Witt-equivalent) to trace forms of field extensions and étale algebras. This line of investigation has been refined by seeking the characterization of forms which are isometric rather than similar to trace forms of field extensions and étale algebras. It turned out that most of the results characterizing quadratic forms up to similarity hold also up to isometry of quadratic forms. The present authors confirm this rule by strengthening an earlier result of A. Prestel. They prove that for a Hilbertian field \(K\) of characteristic zero the trace form of an étale algebra over \(K\) of dimension \(>2\) is isometric to the trace form of some field extension of \(K\). The authors also include several observations on trace forms of étale algebras. Thus they determine forms of this type of dimensions \(<6\) in terms of quadratic form invariants and they relate the existence of trace forms to the field invariants. An open problem asks if every trace form of an étale algebra can be realized as the direct orthogonal sum of trace forms of two-power degree field extensions.
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 12E25
zbMATH DE Number: 737171
zbMATH Keywords: isometry
zbMATH Keywords: quadratic forms
zbMATH Keywords: field extensions
zbMATH Keywords: Hilbertian field
zbMATH Keywords: trace forms of étale algebras
reviewed by: Kazimierz Szymiczek
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 737171