2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/114065For an integer $n>2$, a rank-$n$ matroid is called an $n$-spike if it consists of $n$ three-point lines through a common point such that, for all $k\in\{1, 2, ..., n - 1\}$, the union of every set of $k$ of these lines has rank $k+1$. Spikes are very special and important in matroid theory. In 2003 Wu found the exact numbers of $n$-spikes over fields with 2, 3, 4, 5, 7 elements, and the asymptotic values for larger finite fields. In this paper, we prove that, for each prime number $p$, a $GF(p$) representable $n$-spike $M$ is only representable on fields with characteristic $p$ provided that $n \ge 2p-1$. Moreover, $M$ is uniquely representable over $GF(p)$.8 pagesCombinatoricsNumber Theory05B35; 11B75; 11T99text