TITLE: How do you create projective plane out of a finite field?
QUESTION [21 upvotes]: I have heard and read unclear mentions of links between projective planes and finite fields.
Is it possible to construct a projective plane (or a Steiner system) starting out with a field? Could you, for example, construct the Fano plane with help of a finite field?

REPLY [3 votes]: Does this help?
Let F be a field. We consider the three dimensional vector space over F. Let subspaces of dimension 1 be points. Let subspaces of dimension 2 be lines. The incidence relation is subset inclusion (i.e. if a point is a subset of a line, then you say the point is on the line.)
These definitions satisfy the axioms of a projective plane. You should be able to recover the Fano plane by applying this procedure with the field with two elements.