The 4-rank of $K₂O_F$ for real quadratic fields F

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].…”

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].…”

On the 4-rank of tame kernels of quadratic number fields

“…One can find formulae for the 4-rank of K 2 O F in [3] and [7]. In 1995, Qin [11], [12] gave an efficient method of computing the 4-ranks of K 2 O F . Later Hurrelbrink and Kolster [6] and Qin [13] gave improved methods of computing the 4-ranks.…”