RESEARCH STATEMENT

A finite presemifield is a non-associative division ring. A presemifield possessing a multiplicative identity is a semifield. My research focuses on commutative semifields, which are in essence the closest algebraic structure to a finite field. The only difference is that multiplication in a finite field is associative whereas in a semifield it is not. It is amazing how this subtle difference in multiplication brings about a whole new and complex structure. For exmaple, the classification of finite fields is a well known result; up to isomorphism there is a unique finite field of order p n for every prime p and natural number n. However, it is very clear that there is no such classification of semifields. Three sets associated with a semifields are the left, middle and right nuclei denoted by Nl, Nm and Nr, which are the sets of elements of the semifield that associate on the left, middle or right respectively. These sets are actually fields and serve as an indication of how far from being associative the semifield is. Most of the research concerning semifields so far has had a geometric flavor to it. However, the geometric approach has proved to be particularly successful only in the cas