[Cherubino] Quaderni Elettronici del Seminario di Geometria Combinatoria 2E (Maggio 2001), 1--54
Università degli Studi di Roma "La Sapienza" - Dipartimento di Matematica

Simeon BALL
The geometry of finite fields
(Maggio 2001)

Abstract.These notes are about the geometry of finite fields. The central purpose is to look at the implications that GF(qn) is an n-dimensional vector space over GF(q) to geometries such as PG(n-1,q), AG(n,q) and the classical polar spaces. The very basics of finite geometry have also been included in an attempt to make the notes as self-contained as possible and requiring only some knowledge of linear algebra.

Chapter 1 is a brief introduction to projective spaces and concludes with the basic idea of how to see PG(n-1,q) and AG(n,q) as subsets of elements of the field GF(qn).
Chapter 2 introduces some interesting subsets of points found in projective spaces, in particular maximal arcs in finite projective planes and introduces related incidence structures such as inversive planes and partial geometries. We also prove a theorem about maximal arcs using finite fields.
Chapter 3 is an introduction to polar spaces including representing the classical polar spaces as subsets of finite fields. The section is concluded with the introduction of m-systems of polar spaces and the construction of maximal arcs from particular m-systems.
Chapter 4 contains a very brief introduction to generalised quadrangles and is mainly concerned with ovoids and spreads of the symplectic generalised quadrangle which are again considered as subsets of finite fields.

Author:
Simeon Ball
School of Mathematical Sciences, Queen Mary College, University of London
Mile End Road, London E1 4NS (UK)
e-mail: simeon@maths.qmul.ac.uk
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