Some new classes of division algebras and potential applications to space-time block coding

In this thesis we study some new classes of nonassociative division algebras. First we introduce a generalisation of both associative cyclic algebras and of Waterhouse's nonassociative quaternions. An important aspect of these algebras is the simplicity of their construction, which is a modification of the classical definition of associative cyclic algebras. By taking the parameter used in the classical definition from a larger field, we lose the property of associativity but gain many new examples of division algebras. This idea is also applied to obtain a generalisation of the first Tits construction. We go on to study constructions of Menichetti, Knuth, and Hughes and Kleinfeld, which have previously only been considered over finite fields. We extend these definitions to infinite fields and get new examples of division algebras, including some over the real numbers. Recently, both associative and nonassociative division algebras have been applied to the theory of space-time block coding. We explore this connection and show how the algebras studied in this thesis can be used to construct space-time block codes