Mathematics of Computation

Author(s): Jerzy Browkin; with an appendix by Karim Belabas; Herbert Gangl.
Journal: Math. Comp. 69 (2000), 1667-1683.
MSC (1991): Primary 19C20; Secondary 11R11, 11R70, 11Y40
Posted: March 15, 2000
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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

Modifying the method of Tate, H. Qin has done the same for

and

and M. Ska $\l$ ba for

and

In the present paper we discuss the methods of Qin and Ska $\l$ ba, 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

The results agree with the conjectural structure of $K_{2}\mathcal{O}_{F}$ given in the paper by Browkin and Gangl.

[A1]: S. Arno, The imaginary quadratic fields of class number , Acta Arith. 60 (1992), 321-334.MR 93b:11144
[A2]: S. Arno, M.L. Robinson, F.S. Wheeler, The imaginary quadratic fields with small odd class number, Acta Arith. 83 (1998), 295-330. MR 99a:11123
[BBCO]: C. Batut, D. Bernardi, H. Cohen, M. Olivier, GP/PARI Calculator, version 1.39.
[KH]: K. Belabas, H. Gangl, Generators and relations for $K_2{\mathcal O}_F$ , imaginary quadratic, in preparation.
[BG]: J. Browkin, H. Gangl, Tame and wild kernels of quadratic imaginary number fields, Math. Comp. 68 (1999) 291-305. MR 99c:11144
[BS]: J. Browkin, A. Schinzel, On Sylow -subgroups of $K_2{\mathcal O}_F$ for quadratic number fields , J. Reine Angew. Math. 331 (1982), 104-113.MR 83g:12011
[C]: Harvey Cohn, Advanced Number Theory, Dover Publications, New York, 1962.MR 82b:12001
[Q1]: H. Qin, Computation of $K_{2}{\mathbb Z}[\sqrt {-6}]$ , J. Pure Appl. Algebra 96 (1994), 133-146.MR 95i:11135
[Q2]: -, Computation of $K_{2}{\mathbb Z}\big [\frac{1+\sqrt {-35}}{2}\big ]$ , Chin. Ann. Math. 17B (1) (1996), 63-72. MR 97a:19004
[Q3]: -, Tame kernels and Tate kernels of quadratic number fields, preprint, 1998.
[S]: M. Ska $\l$ ba, Generalization of Thue's theorem and computation of the group $K_2{\mathcal O}_F$ , J. Number Theory 46 (1994), 303-322.MR 95d:19001
[T]: J. Tate, Appendix, Lecture Notes in Math., Vol. 342, Springer Verlag, New York/Berlin, 1973, pp. 429-446. MR 48:3656b
[W]: C. Wagner, Class number and , Math. Comp. 65 (1996), 785-800. MR 96g:11135

Retrieve articles in Mathematics of Computation with MSC (1991): 19C20, 11R11, 11R70, 11Y40

Jerzy Browkin
Affiliation: Institute of Mathematics, University of Warsaw, ul. Banacha 2, PL-02-097 Warsaw, Poland
Email: bro@mimuw.edu.pl

with an appendix by Karim Belabas
Affiliation: Dept. de Mathématiques, Bât. 425, Université Paris-Sud, F-91405 Orsay, France
Email: Karim.Belabas@math.u-psud.fr

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