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For a CM abelian extension $F/K$ of an arbitrary totally real number field$K$, we construct the Stickelberger splitting maps (in the sense of \cite{Ba1})for both the \'etale and the Quillen $K$--theory of $F$ and we use these mapsto construct Euler systems in the even Quillen $K$--theory of $F$. TheStickelberger splitting maps give an immediate proof of the annihilation of thegroups of divisible elements $div K_{2n}(F)_l$ of the even $K$--theory of thetop field by higher Stickelberger elements, for all odd primes $l$. Thisgeneralizes the results of \cite{Ba1}, which only deals with CM abelianextensions of $\Bbb Q$. The techniques involved in constructing our Eulersystems at this level of generality are quite different from those used in\cite{BG1}, where an Euler system in the odd $K$--theory with finitecoefficients of abelian CM extensions of $\Bbb Q$ was given. We work under theassumption that the Iwasawa $\mu$--invariant conjecture holds. This permits usto make use of the recent results of Greither-Popescu \cite{GP} on the \'etaleCoates-Sinnott conjecture for arbitrary abelian extensions of totally realnumber fields, which are conditional upon this assumption. In upcoming work, wewill use the Euler systems constructed in this paper to obtain information onthe groups of divisible elements $div K_{2n}(F)_l$, for all $n>0$ and odd $l$.It is known that the structure of these groups is intimately related to some ofthe deepest unsolved problems in algebraic number theory, e.g. theKummer-Vandiver and Iwasawa conjectures on class groups of cyclotomic fields.We make these connections explicit in the introduction.

The Stickelberger splitting map and Euler systems in the $K$--theory of number fields