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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Quadratic forms and Pfister neighbors in characteristic 2

Authors: Detlev W. Hoffmann and Ahmed Laghribi
Journal: Trans. Amer. Math. Soc. 356 (2004), 4019-4053
MSC (2000): Primary 11E04; Secondary 11E81
Published electronically: February 27, 2004
MathSciNet review: 2058517
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Abstract: We study Pfister neighbors and their characterization over fields of characteristic $2$, where we include the case of singular forms. We give a somewhat simplified proof of a theorem of Fitzgerald which provides a criterion for when a nonsingular quadratic form $q$ is similar to a Pfister form in terms of the hyperbolicity of this form over the function field of a form $\varphi$ which is dominated by $q$. From this, we derive an analogue in characteristic $2$ of a result by Knebusch saying that, in characteristic $\neq 2$, a form is a Pfister neighbor if its anisotropic part over its own function field is defined over the base field. Our result includes certain cases of singular forms, but we also give examples which show that Knebusch’s result generally fails in characteristic $2$ for singular forms. As an application, we characterize certain forms of height $1$ in the sense of Knebusch whose quasi-linear parts are of small dimension. We also develop some of the basics of a theory of totally singular quadratic forms. This is used to give a new interpretation of the notion of the height of a standard splitting tower as introduced by the second author in an earlier paper.

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Additional Information

Detlev W. Hoffmann
Affiliation: Laboratoire de Mathématiques, UMR 6623 du CNRS, Université de Franche-Comté, 16 Route de Gray, F-25030 Besançon Cedex, France
Address at time of publication: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, Great Britain

Ahmed Laghribi
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany

Keywords: Quadratic form, Pfister form, Pfister neighbor, quasi-Pfister form, quasi-Pfister neighbor, function field of a quadratic form, degree of a quadratic form, splitting tower, height of a quadratic form, singular quadratic form
Received by editor(s): January 21, 2003
Received by editor(s) in revised form: June 27, 2003
Published electronically: February 27, 2004
Additional Notes: Both authors have been supported in part by the European research networks FMRX-CT97-0107 and HPRN-CT-2002-00287 “Algebraic $K$-Theory, Linear Algebraic Groups and Related Structures”, and by the program INTAS 99-00817 “Linear Algebraic Groups and Related Linear and Homological Structures”.
Article copyright: © Copyright 2004 American Mathematical Society