The formula of 8-ranks of tame kernels

Abstract

In this paper, we always assume that F = Q( √ d) and E = Q( √ −d), d a squarefree integer, are quadratic number fields with d = u 2 − 2w 2 , u, w ∈ N. This paper is mainly to give the formula: 8-rank of K 2 O F = 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 K 2 O F -8-rank of C(E)| 1. This paper is also to show the relations among {−1, u + √ d} ∈ K 2 O 4 F , the dyadic ideal class of C(E) and the dyadic ideal class of C(E ) for E = Q( √ −2d).