The formula of 8-ranks of tame kernels

In this paper, we always assume that F=Q(d) and E=Q(−d), d a squarefree integer, are quadratic number fields with d=u2−2w2, u,w∈N. This paper is mainly to give the formula: 8-rank of K2OF=8-rank of C(E)+a(F)+σ, where C(E) is the narrow class group of E, a(F)∈{−1,0,1} and σ∈{0,1}; moreover |8-rank of K2OF-8-rank of C(E)|⩽1. This paper is also to show the relations among {−1,u+d}∈K2OF4, the dyadic ideal class of C(E) and the dyadic ideal class of C(E′) for E′=Q(−2d).