# Galois geometry

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Briefly speaking, Galois geometry is analytical and algebraic geometry over a Galois field, that is, geometry over a finite field (cf. also Analytic geometry; Algebraic geometry). Its beginning may be traced back to a result of B. Segre (1954), saying that every \$(q+1)\$-arc, i.e. set of \$q+1\$ three-by-three non-collinear points, of a projective Galois plane \$PG(2, q), q = p^h, p\$ an odd prime number, is an irreducible conic [a7].

The connections between Galois geometry and other branches of mathematics are numerous: classical algebraic geometry and algebra [a18], information theory [a11], physics [a14], coding theory (cf. also Coding and decoding) [a12], cryptography [a17] and mathematical statistics [a11], [a13].

There are graphic characterizations of remarkable algebraic varieties [a15] like quadrics [a10], Veronese varieties, Grassmann varieties, unitals [a8], etc. (cf. also Algebraic variety). Further, there are links between caps and codes [a12], [a13] (cf. also Goppa code), with the classification of \$k\$-sets from the point of view of characters [a9], and with the study of ovals and hyperovals, the theory of spreads and blocking sets and the theory of combinatorial designs [a16] (cf. also Block design).

General references are: [a1], [a2], [a3], [a4], [a5], [a6]. The other references deal with specific topics related with Galois geometry.

How to Cite This Entry:
Galois geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Galois_geometry&oldid=34452
This article was adapted from an original article by M. Scafati Tallini (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article