Primitive and Non Primitive Finite Semifields#

Abstract

A finite semifield (or finite division ring) D is a finite nonassociative ring with identity such that the set is closed under the product and it is a loop. For an arbitrary finite semifield D the structure of its multiplicative loop D* is not known. G.P. Wene introduced the concept of right primitive semifield for those finite semifields D such that D* is equal to the set of principal powers of an element of the semifield, and proved that any semifield of 16, 27, 32, 125 and 343 elements is right primitive. Moreover, he showed that all commutative semifields three-dimensional over a finite field of odd characteristic are right primitive, and conjectured that any finite semifield is right primitive. In this work, we extend his results to any three-dimensional finite semifield over its center, and present a counterexample to the above mentioned conjecture.