Matroids over fp which are rational excluded minors

For a given prime p, we construct a collection of 2^p matroids G_p,a with (1) χ_pf(G_p,a)={p}, and (2) G_p,a is an excluded minor for rational representability. The motivating construction (Section 2) disproves a conjectures of Reid [4], using relatively high-rank, high cardinality matroids. The general construction (Section 3) makes use of ordered partitions (χ_pf(G) denotes the prime-field characteristic set of G, i.e., the set of prime fields over which G may be represented, while G can be represented over fields of no other characteristic.) Finally, Section 4 offers another construction with the same properties–a kind of projective dual to Section 2.