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.