Values at s=-1 of L-functions for multiquadratic extensions of number fields, and annililation of the tame kernel

Suppose that $EE$ is a totally real number field which is the composite of all of its subfields $E$ that are relative quadratic extensions of a base field $F$. For each such $E$ with ring of integers $\O_E$, assume the truth of the Birch-Tate conjecture (which is almost fully established) relating the order of the tame kernel $K_2(\O_E)$ to the value of the Dedekind zeta function of $E$ at $s=-1$, and assume the same for $F$ as well. Excluding a certain rare situation, we prove the annihilation of $K_2(\Oc_EE)$ by a generalized Stickelberger element in the group ring of the Galois group of $EE/F$. Annihilation of the odd part of this group is proved unconditionally.