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.

Key Words:

• Semifield
• Finite division ring
• Right (left) primitive semifield

Acknowledgments

The author wishes to thank professors A. A. Nechaev and V. T. Markov for their guidance during his staying in the Moscow State University in the period April–May 2002, where the results presented in this article were obtained. Also is grateful to professor C. Martínez for her valuable comments for improving the final version of this paper. Partially supported by GE-EXP01-05 and BFM2001-3239-C03-01.

Notes

^#Communicated by I. Shestakov.

^aNevertheless, we shall see that this statement is not exactly true.