On 2-Sylow Subgroups of Tame Kernels

Abstract

Let and be quadratic number fields with 2 ∈ N F/ℚ(F*). Qin (2001, Tables); (2005, Tables), lists the 8-ranks of K 2 O E for all quadratic fields E provided that their discriminants have at most two odd prime divisors, i.e., the 2-ranks of the narrow class groups C(F) are at most 2. In this article, we generalize some of above results. More precisely, we calculate the 2 n -ranks (n ≥ 2) of the tame kernels K 2 O E provided that the 4-ranks of the narrow class groups C(F) are at most 1. We set up relations between the 2 n -ranks (n ≥ 2) of K 2 O E and the 2 m -ranks (m ≥ 2) of C(F). These results are also useful for numerical computations.