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A Note on Hermitian and Quadratic Forms

A Note on Hermitian and Quadratic Forms A NOTE ON HERM1T1AN AND QUADRATIC FORMS D. W. LEWIS Let K be a field (characteristic # 2), and let L be a quadratic extension of K. Then L = K{yja) for some aeK. Let D be the quaternion algebra (a, b/K) generated 2 2 by elements i, j satisfying i = a, j = b, ij = —ji. We will assume that D is a division algebra, i.e. tha t the quadratic form <1> ~ > ~b, ab} is anisotropic. Let — denote the standard involution on L and on D so that y/a = —*Ja on L and I = —i,j= —j on D. We will consider hermitian forms 0 over L and D with respect to the standard involution. Given such a hermitian form </> there is an underlying symmetric bilinear form over K given by •£($ + $) . (Equivalently we have an underlying quadratic form over K given by taking (j)(x x) for all x). Jacobson [1] proved that two hermitian forms over L, or over D, are isometric if and only if their underlying quadratic forms are isometric. We may ask when is a quadratic form over K the underlying form of some http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

A Note on Hermitian and Quadratic Forms

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/11.3.265
Publisher site
See Article on Publisher Site

Abstract

A NOTE ON HERM1T1AN AND QUADRATIC FORMS D. W. LEWIS Let K be a field (characteristic # 2), and let L be a quadratic extension of K. Then L = K{yja) for some aeK. Let D be the quaternion algebra (a, b/K) generated 2 2 by elements i, j satisfying i = a, j = b, ij = —ji. We will assume that D is a division algebra, i.e. tha t the quadratic form <1> ~ > ~b, ab} is anisotropic. Let — denote the standard involution on L and on D so that y/a = —*Ja on L and I = —i,j= —j on D. We will consider hermitian forms 0 over L and D with respect to the standard involution. Given such a hermitian form </> there is an underlying symmetric bilinear form over K given by •£($ + $) . (Equivalently we have an underlying quadratic form over K given by taking (j)(x x) for all x). Jacobson [1] proved that two hermitian forms over L, or over D, are isometric if and only if their underlying quadratic forms are isometric. We may ask when is a quadratic form over K the underlying form of some

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Oct 1, 1979

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