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Quaternary quadratic form

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A quadratic form in four variables. A quaternary quadratic form over a field is related to the algebra of quaternions (cf. Quaternion) over the same field. Namely, corresponding to the algebra with basis , , , and , is the quaternary quadratic form which is the norm of the quaternion,

For quaternary quadratic forms corresponding to quaternion algebras, and only for these, composition of quaternary quadratic forms is defined:

where the coordinates of the vector are bilinear forms in and . Composition of this kind is possible only for quadratic forms in two, four and eight variables.


Comments

The last-mentioned result is known as Hurwitz's theorem; see Quadratic form.

How to Cite This Entry:
Quaternary quadratic form. A.V. Malyshev (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quaternary_quadratic_form&oldid=18432
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098