Cyclotomic and simplicial matroids

Two naturally occurring matroids representable over Q are shown to be dual:the {\it cyclotomic matroid} $\mu_n$ represented by the $n^{th}$ roots of unity$1,\zeta,\zeta^2,...,\zeta^{n-1}$ inside the cyclotomic extension $Q(\zeta)$,and a direct sum of copies of a certain simplicial matroid, consideredoriginally by Bolker in the context of transportation polytopes. A result ofAdin leads to an upper bound for the number of $Q$-bases for $Q(\zeta)$ amongthe $n^{th}$ roots of unity, which is tight if and only if $n$ has at most twoodd prime factors. In addition, we study the Tutte polynomial of $\mu_n$ in thecase that $n$ has two prime factors.