On the 4-rank of the tame kernel K2(O) in positive definite terms

The paper is about the structure of the tame kernel K-2(C) for certain quadratic number fields. There has been recent progress in making explicit the 4-rank of the tame kernel of quadratic number fields and ev en in obtaining results about the 8-rank. The emphasis of this paper is to determine the 4-rank of the tame kernel in definite terms. Our characterizations are in terms of positive definite binary quadratic forms X-2 + 32Y(2), X-2 + 2pY(2), 2X(2) + pY(2) over Z. The results make numerical computations readily available, and the characterizations might generate some interest in "density results" concerning the 4-rank of tame kernels. (C) 2001 Academic Press.