## Matroids Representable over GF(3) and other Fields

#### Abstract

The class of matroids that is representable over $GF(3)$ and some other field
depends on the choice of field.
This paper gives matrix characterisations of these classes.
These characterisations are analogues of the characterisation
of regular matroids as the ones that can be represented over the rationals by
a unimodular matrix. Some consequences of the theory are as follows. A
matroid is representable over $GF(3)$ and $GF(5)$ if and only if it is
representable over $GF(3)$ and the rationals if and only if it is representable
over $GF(p)$ for all odd primes $p$. A matroid is representable over $GF(3)$
and the complex numbers if and only if it is representable over $GF(3)$ and
$GF(7)$. A matroid is representable over $GF(3)$, $GF(4)$ and $GF(5)$ if and
only if it is representable over every field except possible $GF(2)$. If a
matroid is representable over $GF(p)$ for all odd primes $p$, then it is
representable over the rationals.
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