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On the wild kernel of number fields. (Sur le noyau sauvage des corps de nombres.) (French) Zbl 0835.11042

Let \(K\) be an algebraic number field. The wild kernel \(H_2 (K)\) of \(K\) is the subgroup of \(K_2 (K)\), on which all the Hilbert symbols vanish. If \(K\) contains for a given prime number \(l\) all the \(l^r\)-th roots of unity \((r\geq 1)\), then the author describes the quotient \(H_2 (K)/ H_2 (K)^{l^r}\) in terms of a logarithmic divisor class group \(\widetilde {Cl}_K\). The result is analogous to J. Tate’s description of the tame kernel in Invent. Math. 36, 257-274 (1976; Zbl 0359.12011).

MSC:

11R70 \(K\)-theory of global fields
11R29 Class numbers, class groups, discriminants

Citations:

Zbl 0359.12011
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