The Geometry of GF(q 3)

1. Introduction. Inversive geometry involves as basic entities points and circles [2, p. 83; 4, p. 252]. The best known examples of inversive planes (the Miquelian planes) are constructed from a field K which is a quadratic extension of some other field F. Thus the complex numbers yield the Real Inv...

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Bibliographic Details
Published in:Canadian journal of mathematics 1986-06, Vol.38 (3), p.672-696
Main Author: Sherk, F. A.
Format: Article
Language:English
Subjects:
Combinatorics
Combinatorics. Ordered structures
Designs and configurations
Exact sciences and technology
Mathematics
Sciences and techniques of general use
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Summary:1. Introduction. Inversive geometry involves as basic entities points and circles [2, p. 83; 4, p. 252]. The best known examples of inversive planes (the Miquelian planes) are constructed from a field K which is a quadratic extension of some other field F. Thus the complex numbers yield the Real Inversive Plane, while the Galois field GF(q 2)(q = pe , p prime) yields the Miquelian inversive plane M(q) [2, chapter 9; 4, p. 257]. The purpose of this paper is to describe an analogous geometry of M(q) which derives from GF(q 3), the cubic extension of GF(q). The resulting space, is three-dimensional, involving a class of surfaces which include planes, some quadric surfaces, and some cubic surfaces. We explore these surfaces, giving particular attention to the number of points they contain, and their intersections with lines and planes of the space .
ISSN:0008-414X
1496-4279
DOI:10.4153/CJM-1986-035-2