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On the wild kernel of number fields and the group of logarithmic classes. (Sur le noyau sauvage des corps de nombres et le groupe des classes logarithmiques.) (French) Zbl 1009.11062

Let \(K\) be a number field, \(\ell\) a given prime, and \(\widetilde{\mathcal C\ell}_K\) the logarithmic class group of \(K\) introduced by the first author. The basic relation connecting the \(\ell\)-parts of \(\widetilde{\mathcal C\ell}_K\) and the wild kernel of \(K\) is extended and made more precise. Next, the authors prove an analogue of the classical Spiegelungssatz for logarithmic class groups. The last part is devoted to the question whether the triviality of the \(\ell\)-part of \(\widetilde{\mathcal C\ell}_K\) is preserved in a finite Galois \(\ell\)-extension \(L/K\). The result is rather complete if \(K\) is totally real. The main tool is the formula for the central logarithmic classes established by the first author [J. Théor. Nombres Bordx. 6, No. 2, 301–325 (1994; Zbl 0827.11064)]. The formula is extended and a slight error is corrected. The paper contains illustrative numerical examples.

MSC:

11R70 \(K\)-theory of global fields
11R29 Class numbers, class groups, discriminants
19C99 Steinberg groups and \(K_2\)

Citations:

Zbl 0827.11064
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