Quadratic forms, division algebras and inseparable multiquadratic extensions
type: Item
instance of: scholarly article
title: Quadratic forms, division algebras and inseparable multiquadratic extensions
zbMATH Open document ID: 0841.11020
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 1996-07-16T00:00:00Z
full work available at URL: https://eudml.org/doc/227341
review text: For a field \(k\) of characteristic 2 let \(Wq(k)\) be the Witt group of quadratic forms over \(k\). Let \(K\) be an extension field of \(k\) and let \(\iota^*_q: Wq(k)\to Wq(K)\) be the natural morphism. The authors consider the problem of determining the kernel of \(\iota^*_q\) in case of purely inseparable extensions \(K/k\). The case of separable extensions (of characteristic 2) has been investigated earlier, as well as the case of inseparable quadratic extensions [cf. \textit{R. Baeza}, Quadratic forms over semilocal rings, Lect. Notes Math. 655 (Springer Verlag 1978; Zbl 0382.10014)]. When \(K= k( \sqrt{a_1}, \dots, \sqrt{a_n})\) the authors show that \[ \ker \iota^*_q= \langle 1, a_1\rangle Wq(k)+ \cdots+ \langle 1,a_n \rangle Wq(k) \] holds for \(n=2\), and also for all \(n\geq 1\) provided that \(I^2 Wq(k)=0\). A basic observation is that an inseparable quadratic extension \(K/k\) is excellent (in the sense that the anisotropic part of every quadratic form over \(K\) is defined over \(k\)). The paper ends with two results on the structure of division \(k\)-algebras of exponent 2 which contain largely purely inseparable extensions of the ground field.
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 12F15
Mathematics Subject Classification ID: 16K20
zbMATH DE Number: 817249
zbMATH Keywords: division algebras
zbMATH Keywords: Witt group
zbMATH Keywords: quadratic forms
zbMATH Keywords: inseparable extensions
author: Pasquale Mammone
author: Remo Moresi
reviewed by: Kazimierz Szymiczek
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 817249