Editing Finite geometry (section)

===Algebraic construction===
A standard algebraic construction of systems satisfies these axioms.  For a [[division ring]] ''D'' construct an {{nowrap|(''n'' + 1)}}-dimensional vector space over ''D'' (vector space dimension is the number of elements in a basis).  Let ''P'' be the 1-dimensional (single generator) subspaces and ''L'' the 2-dimensional (two independent generators) subspaces (closed under vector addition) of this vector space.  Incidence is containment.  If ''D'' is finite  then it must be a [[finite field]] GF(''q''), since by [[Wedderburn's little theorem]] all finite division rings are fields. In this case, this construction produces a finite projective space. Furthermore, if the geometric dimension of a projective space is at least three then there is a division ring from which the space can be constructed in this manner. Consequently, all finite projective spaces of geometric dimension at least three are defined over finite fields. A finite projective space defined over such a finite field has {{nowrap|''q'' + 1}} points on a line, so the two concepts of order coincide.  Such a finite projective space is denoted by {{nowrap|PG(''n'', ''q'')}}, where PG stands for projective geometry, ''n'' is the geometric dimension of the geometry and ''q'' is the size (order) of the finite field used to construct the geometry.

In general, the number of ''k''-dimensional subspaces of {{nowrap|PG(''n'', ''q'')}} is given by the product:<ref>{{harvnb|Dembowski|1968|loc=p. 28}}, where the formula is given, in terms of vector space dimension, by {{nowrap|N<sub>''k''+1</sub>(''n'' + 1, ''q'')}}.</ref>
:<math>
 {{n+1} \choose {k+1}}_q = \prod_{i=0}^k \frac{q^{n+1-i}-1}{q^{i+1}-1},
</math>
which is a [[Gaussian binomial coefficient]], a ''q'' analogue of a [[binomial coefficient]].