On the 3-rank of tame kernels of certain pure cubic number fields

Abstract

In this paper, we present some explicit formulas for the 3-rank of the tame kernels of certain pure cubic number fields, and give the density results concerning the 3-rank of the tame kernels. Numerical examples are given in Tables 1 and 2. Keywordsthe 3-rank of the tame kernels, pure cubic fields, density MSC(2000): 11R70, 19F15Citation: Li Y Y, Qin H R. On the 3-rank of tame kernels of certain pure cubic number fields.

3-Class groups of cubic cyclic function fields

“…In particular, Gerth studied in great detail the Sylow 3-subgroups of cubic cyclic number fields (see [9]- [10]) and presented analogous results for the 3-rank of the 3-class group of cubic fields to Gauss's for the 2-rank of the 2-class group of quadratic fields. Using Gerth's results on the 3-rank of the class group of cubic number fields, Chen et al [5], Guo [13], Li and Qin [15], and Zhou [23] studied the 3-ranks of tame kernels of cubic cyclic number fields and associated density problems. Recently, in terms of Gerth's results in the number field case, we presented in [22] the function field analogue of the l -rank of class groups of cyclic function fields by the genus theory and Conner-Hurrelbrink exact hexagon for function fields.…”

Odd parts of tame kernels of dihedral extensions

“…Browkin [2] studied tame kernels of cubic cyclic fields with exactly one ramified prime. Li, Qin, Wu and the author obtained some results on tame kernels of cubic and quintic number fields in [10,17,[20][21][22].…”