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The Geometry of GF(q 3)

  • F. A. Sherk (a1)

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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 .

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References

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1. Bruck, R. H., Circle geometry in higher dimensions II, Geom. Ded. 2 (1973), 133188.
2. Coxeter, H. S. M., Introduction to geometry (Wiley, New York, 1969).
3. Coxeter, H. S. M., Projective geometry (Blaisdell, Waltham, Mass., 1964).
4. Dembowski, P., Finite geometries (Springer-Verlag, Berlin, 1968).
5. Hall, M. Jr., Cyclic projective planes, Duke Math. J. 14 (1947), 10791090.
6. Hirschfeld, J. W. P., Projective geometries over finite fields (Clarendon Press, Oxford, 1979).
7. Jungnickel, D. and Vedder, K., On the geometry of planar difference sets, Europ. J. Combinatorics 5 (1984), 143148.
8. Singer, J., A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc. 43 (1938), 377385.
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The Geometry of GF(q 3)

  • F. A. Sherk (a1)

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