The alternative Clifford algebra of a ternary quadratic form

Abstract

We prove that the alternative Clifford algebra of a nondegenerate ternary quadratic form is an octonion ring whose center is the ring of polynomials in one variable over the field of definition.

Keywords:

• Alternative algebras
• Clifford algebras
• octonion algebras
• quadratic forms

2010 MATHEMATICS SUBJECT CLASSIFICATION::

• 17D05
• 11E04
• 11E81
• 15A66

Acknowledgments

We thank Ivan Shestakov and for an insightful discussion and for his careful comments on an earlier version of this paper and an anonymous referee for suggesting significant shortcuts in the proof of the main result.

Additional information

Funding

Uzi Vishne was partially supported by an Israel Science Foundation grant #1623/16.