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Zbl 0822.11079
Thomas, Hervé
Triviality of the 2-rank of the Hilbert kernel. (Trivialité du 2-rang du noyau Hilbertien.) (French)
[J] J. Théor. Nombres Bordx. 6, No.2, 459-483 (1994). ISSN 1246-7405

Let $K$ be an algebraic number field. By a well-known theorem of Matsumoto, the Milnor group $K\sb 2 (K)$ is generated by symbols $(a,b)$, $a,b\in K\setminus \{0\}$, subject to certain relations. These relations are satisfied by the Hilbert symbols $((a,b)/ {\germ p})\sb{\germ p}$, ${\germ p}$ a place of $K$, so that the product of all the Hilbert symbols defines a homomorphism on $K\sb 2 (K)$ whose kernel is the Hilbert or wild kernel $H\sb 2 (K)$; nontrivial elements of $H\sb 2 (K)$ correspond to exotic symbols, that is, symbols which can be detected by global class field theory but not by local class field theory. \par In this paper, the author gives an exhaustive list of the biquadratic fields of the forms $\bbfQ (i, \sqrt{m})$ and $\bbfQ (\sqrt{2}, \sqrt {m})$ for which the 2-rank of the Hilbert kernel is 0. For both types of fields, the results require an analysis of many subcases, which depend on the behaviour of $m$ modulo 8 or 16. The author gives a summary of the ancillary results needed for his calculations and makes a detailed computation in one case. He also discusses the relation between the Hilbert kernel, the regular kernel and the Milnor group $K\sb 2 ({\cal O}\sb K)$ of the ring of integers of $K$, which leads to explicit results on the 2-rank of $K\sb 2 ({\cal O}\sb K)$.
[M.E.Keating (London)]

MSC 2000:: *11R70 K-theory of global fields
11R37 Class field theory for global fields
19F15 Symbols and arithmetic (K-theory)
11R20 Other abelian and metabelian extensions
19C30 $K\sb 2$ and the Brauer group
19C20 Symbols, presentations and stability of K$\sb 2$

Keywords: Milnor $K$-group; Hilbert symbols; class field theory; biquadratic fields; Hilbert kernel; regular kernel

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