We say that a matroid M is representable over a field F if there is some vector space V over F, with some finite set E of vectors of V, so that M is isomorphic to the vectorial matroid of the set E. (Thus we can think of M as just consisting of a bunch of vectors in some vector space over the field F.) We call a matroid that is representable over GF(2) "binary", and over GF(3) "ternary".

It is of tremendous interest and import to classify, for a given matroid M, all the fields over which M is representable, and conversely to see if it's possible to characterize all the matroids that are representable over a given field F. Here are some amazingly impressive and important results for representability:

1. A matroid is GF(2)-representable iff it has no U(2, 4)-minor.
2. A matroid is GF(3)-representable iff it has no minor isomorphic to U(2, 5), U(3, 5), or the Fano plane (i.e., the seven-point projective plane) or its dual.

This sort of theorem, listing all the forbidden minors associated with a specific field, is called an excluded minor characterization.

When I first wrote this page in 1996, there was no other field for which an excluded minor characterization was known. As of this writing, however -- in the summer of 1998 -- an excluded minor characterization for GF(4) has been discovered by Geelen, Gerards and Kapoor. Yet still, it is not yet known whether any other field even has a finite list of excluded minors!

Next, I give the following theorem, which was discovered by Tutte in 1965:

3. A matroid is representable over every field iff it is representable over GF(2) and over some field of characteristic other than two.

A matroid as in this third theorem is called regular (for reasons other than I give here), and it has a vast amount of importance in many areas of mathematics. I won't delve further here.