turkmath.org

Shortly after the classification of finite fields by E. H. Moore, announced in 1893 at the International Mathematical Congress in Chicago, mathematicians turned their attention to algebraic systems (K,+,·), satisfying a slightly weaker set of axioms compared to a finite field. For instance, L. E. Dickson and J. Wedderburn studied finite domains, omitting the axiom: (K^*,·) is commutative. In 1905 they proved that there is no such thing as a non-commutative finite domain. This is now known as Wedderburn’s little theorem. Similarly, Dickson studied algebraic systems (K,+,·) omitting the axiom of associativity (K,+,·) as well: non-associative division algebras. In this case however there do exist proper examples, i.e. which are not fields. This study was continued by his student A. A. Albert, and picked up shortly after by many other mathematicians, including E. Artin, M. A. Zorn, L. A. Skornyakov, R. H. Bruck, D. E. Knuth, E. Kleinfield, I. Kaplansky, to name but a few. Nowadays these algebras are called semifields (a term introduced by D. E. Knuth in 1965), and semifields are studied up to isotopisms instead of isomorphisms. We will explain the BEL-configuration from [1] and report on the structural consequences on the set of isotopism classes of semifields from [2].
[1] S. Ball, G. Ebert, M. Lavrauw. A geometric construction of finite semifields. J. Algebra (2007)
[2] M. Lavrauw and J. Sheekey. On BEL-configurations and finite semifields. arXiv1402.2486