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On the 2-Sylow subgroup of the Hilbert kernel of $$K_ 2$$ of number fields. (English) Zbl 0705.19005
The authors give class field theory interpretations of $$R_ 2(F)/R_ 2(F)^ q$$, where $$R_ 2(F)$$ is the Hilbert kernel in $$K_ 2(F)$$, q a limited power of 2, and F a totally real number field. This interpretation uses a “relative idèle group” I(E/F), where $$E=F(\sqrt{-1})$$, which is such that the $$2^ n$$-ranks of I(E/F) and $$R_ 2(F)$$ are equal as soon as E contains $$2^ n$$-roots of unity; a hypothesis on the Hilbert symbols at 2-adic places is needed and the finiteness of I(E/F) depends on Gross conjecture in E at 2.
This study is made precise for quadratic fields, many examples are given, and some cases of Birch-Tate conjecture are tested; for $$F={\mathbb{Q}}(\sqrt{D})$$, $$D>0$$, a formula for the 2-rank (and the 4-rank) of $$R_ 2(F)$$ is given via the corresponding ranks of the “S-class group” of $$K={\mathbb{Q}}(\sqrt{-D})$$.
Reviewer: G.Gras

##### MSC:
 19F05 Generalized class field theory ($$K$$-theoretic aspects) 11R70 $$K$$-theory of global fields 11R37 Class field theory 11R29 Class numbers, class groups, discriminants 11R11 Quadratic extensions
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