On the 4-rank of the Tame Kernel K₂($\text{O}$) in Positive Definite Terms

Abstract

The paper is about the structure of the tame kernel K₂($\text{O}$) 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 even 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 X²+32Y², X²+2pY², 2X²+pY² over $\text{Z}$. The results make numerical computations readily available, and the characterizations might generate some interest in “density results” concerning the 4-rank of tame kernels.