On the semifield planes of order $5^4$ and dimension 2 over the kernel

Abstract

Enter section title between curly braces In this article we consider the problem of determining all non-Desarguesian semifield planes of order $5^4$ and kernel GF$(5^2)$. We show that the class of $p$-primitive planes is the largest class and besides those the only other semifield planes in the class under study are the generalized twisted field planes. We conjecture that in general these two classes include all the non-Desarguesian semifield planes of order $p^4$ and kernel GF$(p^2)$.