abstract: brumer's conjecture states that stickelberger elements combining values of l-functions at s=0 for an abelian extension of number fields e/f should annihilate the ideal class group of e when it is considered as module over the appropriate group ring. in some cases, an ideal obtained from these stickelberger elements has been shown to equal a fitting ideal connected with the ideal class group. we consider the analog of this at s=-1, in which the class group is replaced by the tame kernel, which we will define. for a field extension of degree 2, we show that there is an exact equality between the fitting ideal of the tame kernel and the most natural higher stickelberger ideal; the 2-part of this equality is conditional on the birch-tate conjecture.