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

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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 Øₑ, assume the truth of the Birch-Tate conjecture (which is almost fully established) relating the order of the tame kernel K₂(Øₑ) 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 K₂(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.