## 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

### Keywords:

wild kernel; logarithmic divisor class group

Zbl 0359.12011
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