G. P. Wene

A survey of finite semifields

Abstract The first part of this paper is a discussion of the known constructions that lead to infinite families of semifields of finite order.The second part is a catalog of the known semifields of finite order. The bibliography includes some references that concern the subject but are not explicitly cited.

On the multiplicative structure of finite division rings

SummaryAll division ringsD of 16, 27, 32, 125 and 343 elements are shown to have aright primitive elementp such that

Idempotent structure of Clifford algebras

\mathcal{D}* = \{ e,ep,(ep)p,...,(...(ep)p...)p\} .

An oval partition of the central units of certain semifield planes

That is, the multiplicative loop is the set of all right multiples of the identitye by the elementp. A construction of Dickson [5] is used to show that all commutative division algebras three-dimensional over a finite field not of characteristic 2 have a primitive element. Examples of division rings of 27, 29 and 211 elements with right primitive elements are given. Finally, a “pre-semifield” is exhibited that does not have a right primitive element.

Albert's construction for semifields of even order

This paper reviews Clifford algebras in mathematics and in theoretical physics. In particular, the little-known differential form realization is constructed in detail for the four-dimensional Minkowski space. This setting is then used to describe spinors as differential forms, and to solve the Klein-Gordon and Kähler-Dirac equations. The approach of this paper, in obtaining the solutions directly in terms of differential forms, is much more elegant and concise than the traditional explicit matrix methods. A theorem given here differentiates between the two real forms of the Dirac algebra by showing that spin can be accommodated in only one of them.

A generalization of the construction of Ilamed and Salingaros

AbstractSpinor spaces can be represented as minimal left ideals of Clifford algebras and they are generated by primitive idempotents. Primitive idempotents of the Clifford algebras Rp, q are shown to be products of mutually nonannihilating commuting idempotent % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqGaaO% qaamaaleaaleaacaaIXaaabaGaaGOmaaaaaaa!3DBD!\[{\textstyle{1 \over 2}}\]2}}\](1+eT), where the k=q−rq−p basis elements eT satisfy eT2=1. The lattice generated by a set of mutually annihilating primitive idempotents is examined. The final result characterizes all Clifford algebras Rp, q with an anti-involution such that each symmetric elements is either a nilpotent or then some right multiple of it is a nonzero symmetric idempotent. This happens when p+q<-3 and (p, q)≠(2, 1).

Semifields antiisomorphic to themselves

A review of the applications of the octonions in physics is given. A construction is presented. Both the Cayley–Dickson algebras and the Clifford algebras arise naturally under this construction from the quaternion algebras. The mathematical properties of the algebras constructed are discussed.

The metric of a space and quadratic algebras

Abstract Let oxy denote a fixed autotopism triangle of a finite commutative semifield plane of even order q N , with middle nucleus GF ( q ). A point I ∉ { x , y , o } is called a central unit of the plane, relative to oxy , if coordinatising the plane by a semifield, in the standard way with I chosen as unit point and ox, oy as axes, yields a commutative semifield D I . It is shown that the set of all central units, relative to a fixed oxy , is partitioned by a set of q − 1 translation (hyper)ovals, any two of which share only the origin o as a common point. The full autotopism group acts transitively on these q − 1 translation ovals, and the ovals, together with the lines ox and oy , define a rational Desarguesian net of degree q .