Quaternion Algebras

Abstract

In this dissertation, we present an analysis of quaternion algebras. We provide a concise overview of the classical theory of quaternion algebras, and move on to study of quaternion algebras along with an additional differential structure. We discuss the classification of first and second order derivations on quaternion algebras and briefly mention the higher order derivations. Furthermore, we discuss about the splitting of differential quaternion algebras and extend these results to quaternion algebras with higher-order ordinary derivations. In the end, we provide an application of quaternion algebras to Number Theory by proving the Lagrange’s four square theorem exploiting Hurwitz factorization within the quaternion ring.