On the 2-primary part of tame kernels of real quadratic fields

Let F=Q(p..), where p = 8t+1 is a prime. In this paper, we prove that a special case of Qin.s conjecture on the possible structure of the 2-primary part of K 2 O F up to 8-rank is a consequence of a conjecture of Cohen and Lagarias on the existence of governing fields. We also characterize the 16-rank of K 2 O F , which is either 0 or 1, in terms of a certain equation between 2-adic Hilbert symbols being satisfied or not.