  In this paper, we construct a higher rank Euler system for the multiplicative
group over a totally real field by using the Iwasawa main conjecture proved by
Wiles. A key ingredient of the construction is to generalize the notion of the
characteristic ideal. Under certain technical assumptions, we prove that all
higher Fitting ideals of a certain $p$-ramified Iwasawa module are described by
analytic invariants canonically associated with Stickelberger elements.
