Elements of order 4 of the Hilbert kernel in quadratic number fields

The formula of 8-ranks of tame kernels

“…Suppose F is an imaginary quadratic field, then take σ = 0. By [1], we know the elements of order at most 2 of K 2 O F : We know that there must be an element α ∈ K 2 O F of order 2 n , n 1, such that the infinite Hilbert symbols η ∞ i (α) = (−1) i , i = 1, 2, see, e.g., [10]. If there are h and ε = −1 satisfying (3.9), (3.9 ) or (3.17), but there is not any β ∈ K 2 O F of order at most 4 such that the infinite Hilbert symbols η ∞ i (β) = (−1) i , i = 1, 2, then, by the construction of …”

“…The tame kernel of F is the Milnor K-group, i.e., K 2 O F . A lot of papers discuss the 2-primary subgroup of K 2 O F in the literature [1,2,4,10,12]. Qin Hourong has supplied an effective method to compute the 4-rank of K 2 O F in [6,7] and the 8-rank of K 2 O F in [8].…”

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…”

Tame Kernels for Biquadratic Number Fields

“…Since F has two dyadic If −d 1 ≡ 1 mod 16, then we assume that −d 1 = u 2 − 2w 2 , u, w ∈ N, w ≡ 4 mod 8, and u ≡ ±1 mod 8 (see [3,14]). Set D 1 = (2,…”