Semifield planes of order p 4 and kernel GF ( p 2 )
Minerva Cordero, Department of Mathematics, The University of Texas at Arlington, Arlington,
Texas 76019, Email address: [email protected]
Abstract:
Let p ≥ 3 be a prime number. It is known that every semifield plane of order p or p 2 is
desarguesian (see [3]) and that every semifield plane of order p 3 is desarguesian or a
twisted field plane (see [4]). The semifield planes of order p 4 and kernel containing
GF ( p 2 ) have been completely determined for p = 3 (see [1]) and p = 5 (see [2]). In
both cases it was found that such semifield planes belong to one of the following classes:
desarguesian, p-primitive or generalized twisted field planes. This led to formulate the
following conjecture in [2]:
Conjecture: Let p ≥ 3 be a prime number. If π is a semifield plane of order p 4 and
kernel containing GF ( p 2 ) , then π is desarguesian, p -primitive, or a generalized
twisted field plane.
In this presentation we show that this conjecture is true for p = 7 and p = 11 .
References:
[1] V. Boerner-Lantz, A new class of semifields, Ph.D. dissertation, Washington State
University, 1983
[2] M. Cordero and R. Figueroa, On the semifield planes of order 54 and dimension 2
over the kernel, Note di Matematica, to appear.
[3] D. Knuth, Finite semifields and projective planes, J. Algebra, 2(1965), 182-217.
[4] G. Menichetti, On a Kaplansky conjecture concerning three-dimensional division
algebras over a finite field, J. Algebra , 47(1977), 400-410.
