On Tame Kernel and Class Group in Terms of Quadratic Forms

Abstract

The paper is to investigate the structure of the tame kernel K 2 O F for certain quadratic number fields F, which extends the scope of Conner and Hurrelbrink ( J. Number Theory 88 (2001), 263–282). We determine the 4-rank and the 8-rank of the tame kernel, the Tate kernel, and the 2-part of the class group. Our characterizations are in terms of binary quadratic forms X 2+32 Y, X 2+64 Y 2, X 2+2 Py 2,2 X 2+ Py 2, X 2−2 Py 2,2 X 2− Py 2. The results are very useful for numerical computations.