ABSTRACT
Recall that a distributive quasifield is called a semifield. Equivalently, a semifield is a ‘non-associative [skew]field’ as seen in the following characterization. The aim of this chapter is to address the following question: what are the possible sizes of finite non-associative semifields? We shall see that semifields that are of order p2 are always fields. Also all translation planes of order 8 are known to be Desarguesian. But the twisted fields of A.A. Albert and the even-order commutative semifields of D.E. Knuth, taken together, demonstrate that for all other primepower orders n at least one non-associative semifield plane of order n exists. The main goal of this chapter is to introduce these planes and demonstrate that they are non-associative.
