Computing the tame kernel of quadratic imaginary fields (with an appendix by Karim Belabas and Herbert Gangl). (English) Zbl 0954.19002
Let \(F=\mathbb Q(\sqrt{-d})\) be an imaginary quadratic number field of discriminant \(-d\) with ring of integers \({\mathcal O}_F\). J. Tate computed the tame kernel \(K_2({\mathcal O}_F)\) for \(d = -3,-4,-7,-8,-11,-15\) [Appendix to H. Bass and J. Tate, “The Milnor ring of a global field”, Lect. Notes Math. 342, 349-446 (1973; Zbl 0299.12013)]. The cases \(d=-24\) and \(d=-35\) were done by Qin using similar methods. M. Skałba improved Tate’s method using a variation of Thue’s theorem to compute the tame kernel for \(d=-19\) and \(d = -20\) [“Generalization of Thue’s theorem and computation of the group \(K_2({\mathcal O}_F)\)”, J. Number Theory 46, No. 3, 303-322 (1994; Zbl 0808.11065)].
In the paper under consideration the authors further improvs the estimates used by Skałba and illustrates his method for the cases \(d = -23\) and \(d = -31\). In an Appendix by K. Belabas and H. Gangl the structure of the tame kernel is listed for all fields \(\mathbb Q(\sqrt{-d})\) with discriminants \(-d\) satisfying \(-151 \leq -d \leq -31\).
In the paper under consideration the authors further improvs the estimates used by Skałba and illustrates his method for the cases \(d = -23\) and \(d = -31\). In an Appendix by K. Belabas and H. Gangl the structure of the tame kernel is listed for all fields \(\mathbb Q(\sqrt{-d})\) with discriminants \(-d\) satisfying \(-151 \leq -d \leq -31\).
Reviewer: M.Kolster (Hamilton/Ontario)
MSC:
| 19C20 | Symbols, presentations and stability of \(K_2\) |
| 11R70 | \(K\)-theory of global fields |
| 11R11 | Quadratic extensions |
| 11Y40 | Algebraic number theory computations |
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