Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Computing the tame kernel of quadratic imaginary fields

Authors: Jerzy Browkin, with an appendix by Karim Belabas and Herbert Gangl
Journal: Math. Comp. 69 (2000), 1667-1683
MSC (1991): Primary 19C20; Secondary 11R11, 11R70, 11Y40
Published electronically: March 15, 2000
MathSciNet review: 1681124
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Abstract: J. Tate has determined the group $K_{2}\mathcal{O}_{F}$ (called the tame kernel) for six quadratic imaginary number fields $F=\mathbb{Q} (\sqrt {d}),$where $d=-3,-4,-7, -8,-11,$ $-15.$ Modifying the method of Tate, H. Qin has done the same for $d=-24$ and $d=-35,$ and M. Ska\lba for $d=-19$ and $d=-20.$

In the present paper we discuss the methods of Qin and Ska\lba, and we apply our results to the field $\mathbb{Q} (\sqrt {-23}).$

In the Appendix at the end of the paper K. Belabas and H. Gangl present the results of their computation of $K_{2}\mathcal{O}_{F}$ for some other values of $d.$The results agree with the conjectural structure of $K_{2}\mathcal{O}_{F}$ given in the paper by Browkin and Gangl.

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Jerzy Browkin
Affiliation: Institute of Mathematics, University of Warsaw, ul. Banacha 2, PL-02-097 Warsaw, Poland

with an appendix by Karim Belabas
Affiliation: Dept. de Mathématiques, Bât. 425, Université Paris-Sud, F-91405 Orsay, France

Herbert Gangl
Affiliation: Max-Planck Institut für Mathematik, Vivatsgaße 7, D-53111, Bonn, Germany

Keywords: Tame kernel, quadratic imaginary fields, Thue's theorem
Received by editor(s): January 14, 1998
Received by editor(s) in revised form: December 7, 1998
Published electronically: March 15, 2000
Article copyright: © Copyright 2000 American Mathematical Society