Galois geometry is the branch of finite geometry

Finite geometry

A finite geometry is any geometric system that has only a finite number of points.Euclidean geometry, for example, is not finite, because a Euclidean line contains infinitely many points, in fact as many points as there are real numbers...

 that is concerned with algebraic

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

 and analytic geometry

Analytic geometry

Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...

 over a finite field (a "Galois" field). More narrowly, a Galois geometry may be defined as a projective space

Projective space

In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

 over a finite field.

Objects of study include vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

s, affine

Affine space

In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...

 and projective space

Projective space

In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

s over finite fields and various structures that are contained in them. In particular, arc

Arc (geometry)

In geometry, an arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, a circular arc is a segment of the circumference of a circle...

s, oval

Oval (projective plane)

In mathematics, an oval in a projective plane is a set of points, no three collinear, such that there is a unique tangent line at each point...

s, hyperovals, unital

Unital

In mathematics, a unital algebra or unitary algebra is an algebra which contains a multiplicative identity element , i.e. an element 1 with the property 1x = x1 = x for all elements x of the algebra....

s, blocking sets, ovoids, caps, spreads and all finite analogues of structures found in non-finite geometries.

External links

• Galois geometry at Encyclopaedia of Mathematics, SpringerLink