45 research outputs found
A geometric characterisation of Desarguesian spreads
We provide a characterisation of -spreads in
that have normal elements in general position. In the same way, we obtain a
geometric characterisation of Desarguesian -spreads in
,
Semifields from skew polynomial rings
Skew polynomial rings were used to construct finite semifields by Petit in
1966, following from a construction of Ore and Jacobson of associative division
algebras. In 1989 Jha and Johnson constructed the so-called cyclic semifields,
obtained using irreducible semilinear transformations. In this work we show
that these two constructions in fact lead to isotopic semifields, show how the
skew polynomial construction can be used to calculate the nuclei more easily,
and provide an upper bound for the number of isotopism classes, improving the
bounds obtained by Kantor and Liebler in 2008 and implicitly in recent work by
Dempwolff
On BEL-configurations and finite semifields
The BEL-construction for finite semifields was introduced in \cite{BEL2007};
a geometric method for constructing semifield spreads, using so-called
BEL-configurations in . In this paper we investigate this construction
in greater detail, and determine an explicit multiplication for the semifield
associated with a BEL-configuration in , extending the results from
\cite{BEL2007}, where this was obtained only for . Given a
BEL-configuration with associated semifields spread , we also show
how to find a BEL-configuration corresponding to the dual spread
. Furthermore, we study the effect of polarities in on
BEL-configurations, leading to a characterisation of BEL-configurations
associated to symplectic semifields.
We give precise conditions for when two BEL-configurations in
define isotopic semifields. We define operations which preserve the BEL
property, and show how non-isotopic semifields can be equivalent under this
operation. We also define an extension of the ```switching'' operation on
BEL-configurations in introduced in \cite{BEL2007}, which, together
with the transpose operation, leads to a group of order acting on
BEL-configurations