On the Classification of Certain Unitary Division Algebras

Nonassociative division algebras play a significant role in Physics and Communications. The finite nonassociative division algebras have a vast range of applications on coding theory, combinatorics and graph theory. This paper deals with a class of finite structures known as division algebras. For a long time division algebras have been studied from a geometric point of view, since they coordinatize certain types of projective planes as an important part of finite geometric incidence. But recent results relating division algebras and coding theory (and also the study of Generalized Galois Rings) have stimulated the study of these rings from a strictly algebraic point of view. This paper follows the second path. Let <i>A</i> be a unital division algebra of order of <i>q<sup>4</sup></i>, q is an odd prime power greater than 3. We assume that <i>A</i> admits an elementary abelian automorphism group <i>E</i> acting freely on <i>A, i.e A≌𝔽<sub>q</sub>[E]</i>. The purpose of this paper is to classify this class of division algebras. In addition, we compute a bound for <i>q</i> and deduce relations among certain structure constants for the quartics associated with <i>A</i>. These relations determine <i>A</i> completely. To achieve these objectives an algebraic geometric approach which is mainly based on the prominent results namely Hasse-Weil theorem and Chevalley-Wraring theorem and the work of Menichetti on n-dimensional algebras over fields of cyclic extensions of degree <i>n</i>.