We give here a sketch of the proof of the following Kaplansky conjecture: any three-dimensional division algebra over a finite field is associative or a twisted field. The detailed proof will appear in a forthcoming paper.
This paper contains the following results on semifields of order , , : a) it is found a class of semifields which generalizes that of R. Sandler [2]; b) it is showed a process which is used for deriving, from any semifield belonging to the class in a), a different semifield with the same order.
Main theorem: In every projective plane of Marshall Hall of order () there are complete q-arcs.
Existence of ovals, which are contained in a class of Moulton's planes of order ( odd), is proved. This class of Moulton's planes is more general than the one considered for the same purpose by Korchmáros in [2]. Furthermore some examples are given of complete -arcs, which are contained in certain Moulton's planes of order ( even).
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