Let \(F\) be an abelian extension of \(\mathbb{Q}\). The Stickelberger theorem constructs an ideal \(S_ 0\) in \(\mathbb{Z}[G(F/\mathbb{Q})]\) that annihilates the class group of \(F\). The ideal \(S_ 0\) is generated by elements of the form \[ \Theta_ 0(b)=(b-(b,F))\sum_{(a;f)=1;1\leq an\) an odd prime, then there is a surjective map \[ \tilde c_{n+1,2}: K_{2n}(F)_ \ell \to H_{cts}^ 2(F,\mathbb{Z}_ \ell(n+1))_ \ell. \] Here \(H_{cts}\) is the continuous cohomology. Theorem C: For an odd, positive integer \(n\) and \(\ell>n\): \[ \left|{{w_{n+1}(F)\zeta_ F(-n)} \over {\prod_{v\mid\ell}w_ n(F_ v)}}\right|_ \ell^{-1}\quad\text{divides}\quad \#(\bigcap_{r\geq 1} K_{2n}(F)^ r)_ \ell. \] Using this result and its consequences, it is possible to determine for each odd prime number \(\ell\), except for irregular prime numbers such that \(\ell\leq n\) and \(\ell\mid nw_{n+1}(\mathbb{Q})\zeta(-n)\), whether or not the short exact sequence \[ 0\to K_{2n}(\mathbb{Z})\to K_{2n}(\mathbb{Q})\to\oplus_ v K_{2n- 1}(k_ v)\to 0 \] splits for the \(\ell\)-torsion part. Theorems A and C give the evidence for the conjecture of Coates and Sinnott.