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APPLICATIONS OF SYSTEMS OF QUADRATIC FORMS TO GENERALISED QUADRATIC FORMS

Part of: Field extensions Forms and linear algebraic groups

Published online by Cambridge University Press:  13 February 2020

A.-H. NOKHODKAR Open the ORCID record for A.-H. NOKHODKAR [Opens in a new window]
A.-H. NOKHODKAR*
Affiliation:
Department of Pure Mathematics, Faculty of Science, University of Kashan, PO Box 87317-51167, Kashan, Iran email a.nokhodkar@kashanu.ac.ir
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Abstract

A system of quadratic forms is associated to every generalised quadratic form over a division algebra with involution of the first kind in characteristic two. It is shown that this system determines the isotropy behaviour and the isometry class of generalised quadratic forms. An application of this construction to the Witt index of generalised quadratic forms is also given.

Keywords

generalised quadratic formsystem of quadratic formsWitt indexdivision algebra with involution

MSC classification

Primary: 11E04: Quadratic forms over general fields
Secondary: 11E39: Bilinear and Hermitian forms 12F10: Separable extensions, Galois theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 3 , December 2020 , pp. 374 - 386
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

This research is partially supported by the University of Kashan under Grant No. 890193/1.

References

Elomary, M. A. and Tignol, J.-P., ‘Classification of quadratic forms over skew fields of characteristic 2’, J. Algebra 240(1) (2001), 366–392.10.1006/jabr.2001.8737CrossRefGoogle Scholar
Jacobson, N., ‘A note on hermitian forms’, Bull. Amer. Math. Soc. 46 (1940), 264–268.10.1090/S0002-9904-1940-07187-3CrossRefGoogle Scholar
Knus, M.-A., Quadratic and Hermitian Forms Over Rings, Grundlehren der mathematischen Wissenschaften, 294 (Springer, Berlin, 1991).CrossRefGoogle Scholar
Knus, M.-A., Merkurjev, A. S., Rost, M. and Tignol, J.-P., ‘The book of involutions’, in: American Mathematical Society Colloquium Publications, 44 (American Mathematical Society, Providence, RI, 1998).Google Scholar
Nokhodkar, A.-H., ‘Hermitian forms and systems of quadratic forms’, Doc. Math. 23 (2018), 747–758.Google Scholar
Nokhodkar, A.-H., ‘Quadratic descent of hermitian forms’, Math. Nachr. 292(10) (2019), 2294–2299.10.1002/mana.201800429CrossRefGoogle Scholar
Pfister, A., Quadratic Forms with Applications to Algebraic Geometry and Topology, London Mathematical Society Lecture Note Series, 217 (Cambridge University Press, Cambridge, 1995).CrossRefGoogle Scholar
Sah, C.-H., ‘A note on hermitian forms over fields of characteristic 2’, Amer. J. Math. 86 (1964), 262–270.10.2307/2373164CrossRefGoogle Scholar
Scharlau, W., Quadratic and Hermitian Forms, Grundlehren der mathematischen Wissenschaften, 270 (Springer, Berlin, 1985).CrossRefGoogle Scholar