On trace forms of Hilbertian fields
type: Item
instance of: scholarly article
title: On trace forms of Hilbertian fields
zbMATH Open document ID: 0662.10015
DOI: 10.1007/BF02566769
author: Martin Krüskemper
author: Winfried Scharlau
published in: Commentarii Mathematici Helvetici
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140124
review text: A quadratic form over a field K is called positive if its signature is nonnegative for every possible ordering of the field K; and it is called algebraic if it is Witt equivalent to the trace form of an algebraic extension L of K. The authors consider the question of when positive forms are algebraic. Suppose K has the property that whenever a polynomial f(T,X) over K in indeterminates T and X is irreducible as a polynomial in X over K(T), then there exists an element \(\beta\) of K such that f(\(\beta\),X) is an irreducible polynomial in X over K. It is proved that the form \(\delta X^ 2\) is algebraic for any element \(\delta\) of K which is positive for every possible ordering of K. This theorem is combined with a theorem by \textit{W. C. Waterhouse} [Arch. Math. 47, 229-231 (1986; Zbl 0607.10013)] to give many special cases in which positive forms are algebraic.
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 12F10
zbMATH DE Number: 4081649
zbMATH Keywords: trace forms
zbMATH Keywords: separable field extension
zbMATH Keywords: Witt equivalence
zbMATH Keywords: Hilbertian field
zbMATH Keywords: Hilbert's irreducibility theorem
zbMATH Keywords: quadratic form
reviewed by: H. F. Kreimer
MaRDI profile type: Publication
OpenAlex ID: W2064565626
description: scientific article; zbMATH DE number 4081649