2-group of positive classes of a number field and wild kernel of \(K\)-theory.
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)].
scientific article; zbMATH DE number 2162036