On Representable Matroids Having Neither U2,5– Nor U3,5–minors

Consider 3–connected matroids that are neither binary nor ternary and have neither U2,5– nor U3,5–minors: for example, AG(3, 2)′, the matroid obtained by relaxing a circuit-hyperplane of AG(3, 2). The main result of the paper shows that no matroid of this sort is representable over any field. This result makes it possible to extend known characterisations of the binary and ternary matroids representable over a field F to ones of the matroids representable over F that have neither U2,5– nor U3,5–minors.