On tame kernel and class group in terms of quadratic forms. (Q1864867)

In [J. Number Theory 88, 263--282 (2001; Zbl 0985.11059)], \textit{P. E. Conner} and \textit{J. Hurrelbrink} characterized the possible \(4\)-ranks of the tame kernels \(K_2(O_F)\) of quadratic number fields \(F\), whose discriminants have exactly two odd prime divisors \(p \equiv -1 \bmod 8\), \(l \equiv 1 \bmod 8\), satisfying \((\frac{p}{l}) = 1\), in terms of positive definite quadratic forms. In the paper under consideration the author treats the other two interesting cases of two odd discriminant divisors: \(p \equiv l \equiv 1 \bmod 8\) and \(p \equiv l \equiv -1 \bmod 8\), and at the same time extends the results to include information about the \(8\)-rank of the tame kernel, and the relations to the structure of the \(2\)-part of the class groups.