Cyclotomic and Simplicial Matroids

We show that two naturally occurring matroids representable over Q are equal: the cyclotomic matroid μn represented by the n roots of unity 1, ζ, ζ, . . . , ζ inside the cyclotomic extension Q(ζ), and a direct sum of copies of a certain simplicial matroid, considered originally by Bolker in the context of transportation polytopes. A result of Adin leads to an upper bound for the number of Q-bases for Q(ζ) among the n roots of unity, which is tight if and only if n has at most two odd prime factors. In addition, we study the Tutte polynomial of μn in the case that n has two prime factors.