A family of semifields in characteristic 2

TL;DR

This paper introduces a new family of semifields in characteristic 2, constructed via projective polynomials, revealing their properties, classifications, and automorphism groups, and expanding the taxonomy of such algebraic structures.

Contribution

It constructs and characterizes a novel family of semifields in characteristic 2, including their nuclei, isotopy classes, and automorphism groups, and completes the taxonomy of quadratic semifields.

Findings

Non-degenerate semifields have nuclei matching the center.

They are not isotopic or Knuth equivalent to commutative semifields.

Complete classification of certain characteristic 2 semifields.

Abstract

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 nucleus. The non-degenerate members of our family display a very different behaviour. Their left and right nucleus agrees 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 nucleus and not isotopic to twisted fields. This includes {determining} when two such semifields are isotopic and the order of the autotopism group.