A finite presemifield of order q = p^r (p a prime) is an algebra (F,+,∗) of order q which satisfies the axioms of the field of order q with the possible exception of the associativity of multiplication and the existence of an identity element of multiplication. A presemifield is a semifield if in addition an identity element of multiplication exists. The addition in a presemifield may be identified with the addition in the field of the same order. A presemifield is commutative if its multiplication is commutative. A geometric motivation to study (pre)semifields comes from the fact that there is a bijection between presemifields and projective planes of the same order which are translation planes and also duals of translation planes. Presemifields (F,+,∗) and (F,+,◦) of order q = pr are defined to be isotopic if there exist elements α1,α2,β ∈ GL(r, p) such that β(α1(x) ∗ α2(y)) = x ◦ y always holds. This equivalence relation is motivated by the geometric link as well.
In [2017/1] we construct and describe the basic properties of a family of semifields in characteristic 2. The construction relies on the properties of projective polynomials over finite fields. We start by associating non-associative products to each such polynomial. The resulting presemifields form the degenerate case of our family. They are isotopic to the Knuth semifields which are quadratic over left and right nuclei. The non-degenerate members of our family display a very different behavior. Their left and right nuclei agree with the center, the middle nucleus is quadratic over the center. None of those semifields is isotopic or Knuth equivalent to a commutative semifield. As a by-product we obtain the complete taxonomy of the characteristic 2 semifields which are quadratic over the middle nucleus, bi-quadratic over the left and right nuclei and not isotopic to twisted fields. This includes determining when two such semifields are isotopic and the order of the autotopism group.
In [2018/4] we study a large family of semifields in odd characteristic, which contains the commutative Budaghyan–Helleseth semifields as well as semifields which are not isotopic to commutative semifields. Using a large group of autotopisms we obtain a complete classi- fication result in certain parametric subcases.