Values at s = −1 of L ‐functions for multi‐quadratic extensions of number fields and annihilation of the tame kernel

Suppose that ℰ 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 a ring of integers E , assume the truth of the 2‐primary part of the Birch–Tate conjecture relating the order of the tame kernel K 2 ( 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 ( ℰ ) by a generalized Stickelberger element in the group ring of the Galois group of ℰ/ F . Annihilation of the odd part of this group is proved unconditionally. This result on the odd part establishes a special case of a conjecture stated by Snaith.