  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, 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 $Q(\zeta)$ among
the $n^{th}$ 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 $\mu_n$ in the
case that $n$ has two prime factors.
