  In this paper, we construct (many) Kolyvagin systems out of Stickelberger
elements, utilizing ideas borrowed from our previous work on Kolyvagin systems
of Rubin-Stark elements. We show how to apply this construction to prove
results on the odd parts of the ideal class groups of CM fields which are
abelian over a totally real field, and deduce the main conjecture of Iwasawa
theory for totally real fields (for totally odd characters). Although the main
results of this paper have already been established by Wiles, our approach
provides another example (which slightly differs from the case of Stark
elements) on how to study Kolyvagin systems of core rank r > 1 (in the sense of
Mazur and Rubin). Also, by making use of the 'rigidity' of the collection of
Kolyvagin systems, we establish a link between the Stickelberger elements and
the Rubin-Stark elements.

Number Theory