A unifying construction of semifields of order $p^{2m}$

TL;DR

This paper introduces two new, unifying constructions for semifields of order p^{2m} that encompass and extend many existing constructions, providing a comprehensive framework and identifying numerous new semifields.

Contribution

It presents two novel constructions that unify and generalize existing semifield constructions, including those by Dickson and Taniguchi, and identifies conditions for equivalence and novelty.

Findings

Unifies over a dozen semifield constructions

Provides conditions for semifield equivalence

Constructs many new inequivalent semifields

Abstract

In this article, we present two new constructions for semifields of order $p^{2m}$. Together, the constructions unify and generalize around a dozen distinct semifield constructions, including both the oldest known construction by Dickson and the largest known construction in odd characteristic by Taniguchi. The constructions also provably yield many new semifields. We give precise conditions when the new semifields we find are equivalent and count precisely how many new inequivalent semifields we construct.