On 14-dimensional quadratic forms in \(I^3\), 8-dimensional forms in \(I^2\), and the common value property
type: Item
instance of: scholarly article
title: On 14-dimensional quadratic forms in \(I^3\), 8-dimensional forms in \(I^2\), and the common value property
zbMATH Open document ID: 0914.11014
author: Detlev Hoffmann
author: Jean-Pierre Tignol
published in: Documenta Mathematica
publication date: 1998-10-22T00:00:00Z
full work available at URL: https://eudml.org/doc/223375
full work available at URL: 06.html
review text: The authors study several properties of fields expressed in terms of the behavior of higher-dimensional quadratic forms. The difference of two 3-fold Pfister forms over a field \(F\) belongs to \(I^3(F)\), the third power of the fundamental ideal of the Witt ring \(W(F)\), and the authors say a field \(F\) satisfies \(D(14)\) if each 14-dimensional quadratic form in \(I^3(F)\) is similar to the difference of two 3-fold Pfister forms. The property \(D(8)\) requires that each form in \(I^2(F)\) of dimension 8 with Clifford invariant represented by a biquaternion algebra is the sum of two forms similar to 2-fold Pfister forms, which reverses another generally valid implication. \(D(4)\) holds for \(F\) when, for every pair of 4-dimensional quadratic forms with the same signed determinant, if the forms become equivalent over a quadratic extension field of \(F\), then they contain subforms similar over \(F\). Finally, \(D(2)\) is a common value property studied earlier in the literature: if two binary forms over \(F\) represent a common value over a quadratic extension \(E\) of \(F\), then they represent over \(E\) a common value in \(F\). The authors establish that \(D(2)\) implies \(D(4)\), and also \(D(8)\) implies \(D(14)\). They show that the implications cannot be reversed in general. It is not known whether \(D(4)\) implies \(D(8)\). The authors study in detail the behavior of the properties \(D(n)\) when passing from a field \(K\) with a 2-henselian discrete valuation to the residue class field \(k\), or from \(k\) to \(K\). For example, they establish that if \(D(2n)\) for \(n=2,4\) (or \(D(14)\)) holds for \(K\), then \(D(n)\) (or \(D(8)\), resp.) holds for \(k\). Also \(D(2)\) lifts from \(k\) to \(K\), \(D(4)\) and \(D(8)\) for \(k\) imply \(D(8)\) for \(K\), and \(D(8)\) for \(k\) implies \(D(14)\) for \(K\). These results are then used to give a series of new examples of fields having or failing to have the properties \(D(n)\). The fields of (iterated) form…
reviewed by: Kazimierz Szymiczek
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 16K20
zbMATH DE Number: 1213152
zbMATH Keywords: Clifford invariant
zbMATH Keywords: common value property
zbMATH Keywords: common slots of quaternion algebras
zbMATH Keywords: biquaternion algebras
zbMATH Keywords: Hasse number
zbMATH Keywords: higher-dimensional quadratic forms
zbMATH Keywords: fundamental ideal of the Witt ring
zbMATH Keywords: Pfister forms
zbMATH Keywords: binary forms
zbMATH Keywords: fields of formal power series
zbMATH Keywords: excellence property of an extension field
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1213152