The 2-Sylow subgroups of the tame kernel of imaginary quadratic fields

On Tame Kernel and Class Group in Terms of Quadratic Forms

“…Qin got the conditions of K 2 O F with elements of order 4 in [10,11], which induces to get the 4-rank of K 2 O F by computing the rank of R' e edei matrix and solutions of some systems of linear equations (see [7,15,16]). In [10,11,14] In [5], they also get the 4-rank of K 2 O F in terms of the above positive definite binary forms for…”

“…Let O F be the ring of integers of an algebraic number field F : The tame kernel of O F is the Milnor K-group K 2 O F : The paper is to characterize the 4-rank and the 8-rank of the tame kernel, the Tate kernel, and the class group for quadratic number fields whose discriminants have exactly two odd prime divisors p; l; p l AE1 mod 8 with ð l p Þ ¼ 1: A lot of people investigate the structure of the 2-Sylow subgroup of K 2 O F : For quadratic number fields F ; we know 2-ranks, 4-ranks, and 8-ranks of K 2 O F by the solution of indefinite norm equations in [2,7,10,11,12,16], and we also know the relation between two 2-Sylow subgroups of the tame kernel and the narrow class group in [15,16].…”

Tame Kernels for Biquadratic Number Fields

“…In this paper, we express the forms of all elements of order at most 4 in K 2 O E by various F 2 -matrices consisting of Legendre symbols and local Hilbert symbols analogous to Rédei matrix, but it has to be pointed out that this paper uses the Hurrelbrink and Kolster's method of relative quadratic extension [5] to investigate specially biquadratic number fields E and generalizes the results of Browkin, Schinzel [1] and Qin [9,10].…”

On two-primary algebraic K-theory of quadratic number rings with focus on K₂

“…This connection will be explained in the calculations. To the authors knowledge, the latest and most complete calculation of K 2 of quadratic number rings can be found in [13] and [14]. We would also like to mention the recent papers [18] and [19].…”