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The Stickelberger splitting map in the case of abelian extensions $F / \Q$was defined in [Ba1, Chap. IV]. The construction used Stickelebrger's theorem.For abelian extensions $F / K$ with an arbitrary totally real base field $K$the construction of \cite{Ba1} cannot be generalized since Brumer's conjecture(the analogue of Stickelberger's theorem) is not proved yet at that level ofgenerality. In this paper, we construct a general Stickelberger splitting mapunder the assumption that the first Stickelberger elements annihilate theQuillen $K$--groups groups $K_2 ({\mathcal O}_{F_{l^k}})$ for the Iwasawa tower$F_{l^k} := F(\mu_{l^k})$, for $k \geq 1.$ The results of [Po] give examples ofCM abelian extensions $F/K$ of general totally real base-fields $K$ for whichthe first Stickelberger elements annihilate $K_2 ({\mathcal O}_{F_{l^k}})_l$for all $k \geq 1$, while this is proved in full generality in [GP], under theassumption that the Iwasawa $\mu$--invariant $\mu_{F,l}$ vanishes. As aconsequence, our Stickelberger splitting map leads to annihilation results aspredicted by the original Coates-Sinnott conjecture for the subgroups$div(K_{2n}(F)_l)$ of $K_{2n}(O_F)_l$ consisting of all the $l$--divisibleelements in the even Quillen $K$-groups of $F$, for all odd primes $l$ and all$n$. } In \S6, we construct a Stickelberger splitting map for \'etale$K$--theory. Finally, we construct both the Quillen and \'etale Stickelbergersplitting maps under the more general assumption that for some arbitrary butfixed natural number $m>0$, the corresponding $m$--th Stickelberger elementsannihilate $K_{2m} ({\mathcal O}_{F_k})_l$ (respectively $K^{et}_{2m}({\mathcal O}_{F_k})_l$), for all $k$

On the Stickelberger splitting map in the $K$--theory of number fields