Clifford algebras of binary homogeneous forms

Abstract

We study the generalized Clifford algebras associated to homogeneous binary forms of prime degree p, focusing on exponentiation forms of p-central spaces in division algebra.For a two-dimensional p-central space, we make the simplifying assumption that one basis element is a sum of two eigenvectors with respect to conjugation by the other. If the product of the eigenvalues is 1 then the Clifford algebra is a symbol Azumaya algebra of degree p, generalizing the theory developed for p=3. Furthermore, when p=5 and the product is not 1, we show that any quotient division algebra of the Clifford algebra is a cyclic algebra or a tensor product of two cyclic algebras, and every product of two cyclic algebras can be obtained as a quotient. Explicit presentation is given to the Clifford algebra when the form is diagonal.