There are exactly seven excluded minors for the class of GF(4)--representable matroids. 1 Introduction We prove the following theorem. Theorem 1.1 A matroid M is GF(4)--representable if and only if M has no minor isomorphic to any of U 2;6 , U 4;6 , P 6 , F \Gamma 7 , F \Gamma 7 , P 8 , and P 00 8 . The definitions for these matroids, with a summary of their interesting properties, can be found in the Appendix. Other than P 00 8 , they were all known to be excluded minors for GF(4)-- representability (see Oxley [13,15]). The matroid P 00 8 is obtained by relaxing the unique pair of disjoint circuit--hyperplanes of P 8 . Ever since Whitney's introductory paper [24] on matroid theory, researchers have sought ways to distinguish the representable matroids. For any field F, the class of F--representable matroids is closed under taking minors. Thus, it is natural to characterize the minor--minimal matroids that are not F--representable; we refer to such matroids as excluded ...