Computing the tame kernel of quadratic imaginary fields

Browkin, Jerzy; Belabas, with an appendix by Karim; Gangl, Herbert

doi:10.1090/S0025-5718-00-01182-0

Computing the tame kernel of quadratic imaginary fields
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by Jerzy Browkin, with an appendix by Karim Belabas and Herbert Gangl PDF

Math. Comp. 69 (2000), 1667-1683 Request permission

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. Skałba for $d=-19$ and $d=-20.$ In the present paper we discuss the methods of Qin and Skałba, 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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