A large number of papers have contributed to determining the structure of the tame kernel $K{₂}_F$ of algebraic number fields F. Recently, for quadratic number fields F whose discriminants have at most three odd prime divisors, 4-rank formulas for $K{₂}_F$ have been made very explicit by Qin Hourong in terms of the indefinite quadratic form x² - 2y² (see [7], [8]). We have made a successful effort, for quadratic number fields F = ℚ (√(±p₁p₂)), to characterize in terms of positive definite binary quadratic forms,...

1. Introduction. For quadratic fields whose discriminant has few prime divisors, there are explicit formulas for the 4-rank of $K{₂}_E$. For quadratic fields whose discriminant has arbitrarily many prime divisors, the formulas are less explicit. In this paper we will study fields of the form $\mathbb{Q}\left(\surd (p₁...p_k)\right)$, where the primes $p_i$ are all congruent to 1 mod 8. We will prove a theorem conjectured by Conner and Hurrelbrink which examines under what conditions the 4-rank of $K{₂}_E$ is zero for such fields. In the course of proving...

We give exhaustive list of biquadratic fields $K=\mathbb{Q}(i,\sqrt{m})$ and $K=\mathbb{Q}(\sqrt{2},\sqrt{m})$ without $2$-exotic symbol, i.e. for which the $2$-rank of the Hilbert kernel (or wild kernel) is zero. Such $K=\mathbb{Q}(i,\sqrt{m})$ are logarithmic principals [J3]. We detail an exemple of this technical numerical exploration and quote the family of theories and results we utilize. The $2$-rank of tame, regular and wild kernel of $K$-theory are connected with local and global problem of embedding in a $Z_2$-extension. Global class field theory can describe the $2$-rank of the Hilbert...