PLANAR POLYNOMIALS FOR COMMUTATIVE SEMIFIELDS WITH SPECIFIED NUCLEI
ROBERT S. COULTER, MARIE HENDERSON, AND PAMELA KOSICK
Abstract. We consider the implications of the equivalence of commutative semifields of odd order and planar Dembowski-Ostrom polynomials. This equivalence was outlined recently by Coulter and Henderson. In particular, following a more general statement concerning semifields we identify a form of planar Dembowski-Ostrom polynomial which must define a commutative semifield with the nuclei specified. Since any strong isotopy class of commuta- tive semifields must contain at least one example of a commutative semifield described by such a planar polynomial, to classify commutative semifields it is enough to classify planar Dembowski-Ostrom polynomials of this form and determine when they describe non-isotopic commutative semifields. We prove several results along these lines. We end by introducing a new commutative semifield of order 38 with left nucleus of order 3 and middle nucleus of order 32.
1. Introduction A finite semifield R is a ring with no zero-divisors, a multiplicative identity and left and right distributivity. If we do not insist on the existence of a multiplicative identity, then we call the ring a presemifield. It is not assumed that R is commu- tative or associative. Though the definition extends to infinite objects, this article is only concerned with the finite case. The additive group of a semifield must be elementary abelian and thus the order of any semifield is necessarily a prime power; for a simple proof see Knuth [12, Section 2.4]. Finite fields satisfy these requirements and so the existence of semifields is clear. Those semifields which are not fields are called proper semifields. The first proper semifields identified were the commutative semifields of Dickson [7] which have order q2 with q an odd prime power. Dickson may have been led to study semifields following the publication of Wedderburn’s Theorem [20], which appeared the year before [7] and which Dickson was the first person to provide a correct proof for (see Parshall [18]). As no new structures are obtained by removing commutativity, it is reasonable to investigate those structures which are non-associative instead. The role of semifields in projective geometry was confirmed following the intro- duction of coordinates in non-Desarguesian planes by Hall [8]. Subsequent to Hall’s work, Lenz [13] developed and Barlotti [1] refined what is now known as the Lenz- Barlotti classification, under which semifields correspond to projective planes of Lenz-Barlotti type V.1. In some sense, modern interest in semifields can be traced back to the important work of Knuth [12]. Presently, semifields are enjoying a true
Date stamped: September 10, 2007 The first author was partially supported by a University of Delaware Research Foundation Grant. He also gratefully acknowledges the support of the Victoria University of Wellington, New Zealand, where part of this research was conducted during a sabbatical in the second half of 2005. 1 Des. 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