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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