2-group of positive classes of a number field and wild kernel of \(K\)-theory.
type: Item
instance of: scholarly article
title: 2-group of positive classes of a number field and wild kernel of \(K\)-theory.
zbMATH Open document ID: 1075.11076
author: Jean-François Jaulent
author: Florence Soriano-Gafiuk
published in: Journal of Number Theory
publication date: 2005-04-28T00:00:00Z
review text: For an odd prime number \(\ell\) and a number field \(F\) containing \(\mu_{\ell},\) the wild kernel \(W\, K_{2}(F)\) reduced modulo \(\ell\) satisfies a canonical isomorphism \(W\, K_{2}(F)/W\, K_{2}(F)^{\ell} \simeq \mu_{\ell} \otimes \widetilde{C\!\ell}_{F},\)where \(\widetilde{C\!\ell}_{F}\) is an Iwasawa descent module [\textit{P. Scheider}, Math. Z. 168, 181--205 (1979; Zbl 0421.12024)] which also admits a description in terms of certain ``logarithmic'' functions, hence the terminology of ``logarithmic classes'' [\textit{J.-F. Jaulent}, Acta Arith. 67, 335--348 (1994; Zbl 0835.11042)]. If \(\ell = 2,\) and \(F\) does not contain \(\mu_{4},\) the above isomorphism can fail. In this paper, the authors introduce ``positive'' versions of the logarithmic classes in order to recover ad hoc isomorphisms. This yields a criterion for the triviality of the 2-part of the wild kernel. In the particular case of quadratic fields, this leads to a ``logarithmic'' interpretation of diophantine conditions previously obtained by \textit{J. Browkin} and \textit{A. Schinzel} [J. Reine. Angew. Math. 331, 104--113 (1982; Zbl 0493.12013)].
reviewed by: Thong Nguyen Quang Do
Mathematics Subject Classification ID: 11R70
zbMATH DE Number: 2162036
zbMATH Keywords: wild kernel
zbMATH Keywords: logarithmic classes
MaRDI profile type: Publication
full work available at URL: j.jnt.2003.09.012
OpenAlex ID: W1971415734
cites work: Genus theory of global fields
cites work: On Sylow 2-subgroups of K2OF for quadratic number fields F.
cites work: On the 2-Sylow subgroup of the Hilbert kernel of $K_{2}$ of number fields
cites work: Generalized invariant classes
cites work: On regular number fields
cites work: The 2-Sylow subgroup of the wild kernel of exceptional number fields
cites work: Sur le noyau sauvage des corps de nombres
cites work: Logarithmic classes of number fields
cites work: Global \(\ell\)-adique clean field theory
cites work: Signed logarithmic classes of number fields
cites work: \(p\)-rational fields, \(p\)-regular fields and restricted ramification
cites work: On the wild kernel of number fields and the group of logarithmic classes
cites work: Quadratic number fields having \([-1,-1]\) as a non-trivial element of their wild kernel
cites work: Classes logarithmiques ambiges des corps quadratiques
cites work: Logarithmic classes in the narrow sense
cites work: Relations between \(K_2\) and Galois cohomology
DOI: 10.1016/J.JNT.2003.09.012
description: scientific article; zbMATH DE number 2162036