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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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