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Signed logarithmic classes of number fields. (Classes logarithmiques signées des corps de nombres.) (French) Zbl 0997.11095
The author defines a signed logarithmic 2-class group of a number field as the actual analogue in the logarithmic context of the usual restricted ideal 2-class group.
Logarithmic classes were introduced by the author [J. Théor. Nombres Bordx. 6, 301-325 (1994; Zbl 0827.11064)] in connection with the wild kernel of $$K$$-theory: as a matter of fact, when the number field $$K$$ contains the $$2p$$th roots of unity for an odd prime $$p$$, there exists a canonical isomorphism $$\mu_p \otimes\widetilde{\text{Cl}}_K\simeq WK_2(K)/ W(K_2(K)^p$$ between the twisted logarithmic $$p$$-class group and the $$p$$-quotient of the wild kernel [Acta Arith. 67, 335-348 (1994; Zbl 0835.11042)]. And since the logarithmic class groups are easily computable, as explained by F. Diaz y Diaz and the reviewer [J. Number Theory 76, 1-15 (1999; Zbl 0930.11079)], this gives rise to an algorithmic approach to the Hilbert kernel. For $$p=2$$ however, the isomorphism above is not valid without the assumption $$i\in K$$ (in fact when $$i$$ is not contained in the cyclotomic $$\mathbb{Z}_2$$-extension $$K^c$$ of $$K$$) and the situation is more intricate.
In the article under review, the author defines a logarithmic signature that involves both the real places and certain 2-adic ones (namely the 2-adic places $${\mathfrak p}$$ such that the cyclotomic $$\mathbb{Z}_2$$-extension $$K_{\mathfrak p}^c$$ of the local field $$K_{\mathfrak p}$$ does not contain the 4th root of unity $$i$$). By this way he extends a previous generalization of the reviewer [Manuscr. Math. 93, 409-420 (1997; Zbl 0887.11044)]. Unfortunately there is a misprint in Lemma 8 that affects some formulas in Section 3 of a factor equal to 1 or 2. This point will be corrected in a Corrigendum to appear. Finally applications to the wild kernel will also follow in a forthcoming article.

##### MSC:
 11R29 Class numbers, class groups, discriminants 11R70 $$K$$-theory of global fields 11R99 Algebraic number theory: global fields
##### Keywords:
signed logarithmic 2-class group; wild kernel
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##### References:
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