Galois geometry

The Fano plane, the projective plane over the field with two elements, is one of the simplest objects in Galois geometry.

Galois geometry (so named after the 19th century French Mathematician Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or Galois field).[1] More narrowly, a Galois geometry may be defined as a projective space over a finite field.[2]

Introduction

Objects of study include vector spaces, affine and projective spaces over finite fields and various structures that are contained in them. In particular, arcs, ovals, hyperovals, unitals, blocking sets, ovoids, caps, spreads and all finite analogues of structures found in non-finite geometries.

George Conwell gave an early demonstration of Galois geometry in 1910 when he characterized a solution of Kirkman's schoolgirl problem as a partition of sets of skew lines in PG(3,2), the three-dimensional projective geometry over the Galois field GF(2).[3] Similar to methods of line geometry in space over a field of characteristic 0, Conwell used Plücker coordinates in PG(5,2) and identified the points representing lines in PG(3,2) as those on the Klein quadric.

In 1955 Beniamino Segre characterized the ovals for q odd. Segre's theorem states that in a Galois geometry of odd order (a projective plane defined over a finite field of odd characteristic) every oval is a conic. At the 1958 International Mathematical Congress Segre presented a survey of results in Galois geometry known up to then.[4]

See also

Notes

  1. ↑ SpringerLink
  2. ↑ "Projective spaces over a finite field, otherwise known as Galois geometries, ...", (Hirschfeld & Thas 1992)
  3. ↑ George M. Conwell (1910) "The 3-space PG(3,2) and its Groups", Annals of Mathematics 11:60–76 doi:10.2307/1967582
  4. ↑ Segre, Beniamino (1958), On Galois Geometries (PDF), International Mathematical Union

References

  • Hirschfeld, J. W. P. (1979), Projective Geometries Over Finite Fields, Oxford University Press, ISBN 978-0-19-850295-1, emphasizing dimensions one and two.
  • Hirschfeld, J. W. P. (1985), Finite Projective Spaces of Three Dimensions, Oxford University Press, ISBN 0-19-853536-8, dimension 3.
  • Hirschfeld, J. W. P.; Thas, J. A. (1992), General Galois Geometries, Oxford University Press, ISBN 978-0-19-853537-9, treating general dimension.
  • De Beule, Jan; Storme, Leo (2011), Current Research Topics in Galois Geometry, Nova Science Publishers, ISBN 978-1-61209-523-3


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