The tame kernels of number fields (Q2900291)

This is a survey of results obtained by numerous authors on the \(p^n\)-rank of \(K_2 {\mathcal O}_F,\) \({\mathcal O}_F\) being the ring of integers of a number field \(F\). For \(p = 2\), \(n = 1, 2, 3\), the authors concentrate on quadratic fields. Results for \(p\) odd are more scarce and concern particular fields (quadratic, sometimes cyclic cubic or quintic fields). In spite of the number and variety of the papers cited here, one can draw a guideline in the methods which are most widely used : \(K_2 {\mathcal O}_F\) is identified with the tame kernel in \(K_2 F\) ; apart from computation of symbols, the main tools are reflection inequalities in the style of Scholz, relating the \(p\)-ranks of \(K_2 {\mathcal O}_F\) and of the class group of \(F(\mu_p)\) (see e.g., [\textit{H. Qin}, J. Pure Appl. Algebra 214, No. 7, 1181--1192 (2010; Zbl 1223.11136)], where \(F\) is a quadratic extension of a totally real field) ; when computing numerical tables, one can appeal to the Lichtenbaum conjecture (now a theorem for abelian fields) to bound the order of \(K_2 {\mathcal O}_F.\) But note that the unifying point of view given by Iwasawa theory and the Quillen-Lichtenbaum conjecture (when of course it can be exploited) is rarely used : namely, the \(p\)-primary part of the wild kernel \(WK_{2m-2}(F)\) is canonically isomorphic to the co-invariants of the \((-m)\)-twist of the unramified and totally split Iwasawa module attached to the cyclotomic \({\mathbb Z}_p\)-extension of \(F(\mu_p)\) [\textit{P. Schneider}, Math. Z. 168, 181--205 (1979; Zbl 0421.12024)]. For applications, see e.g., [\textit{M. Kolster}, Prog. Math. 116, 37--62 (1994; Zbl 1043.19500)] or \textit{T. Nguyen Quang Do} [Théorie d'Iwasawa des noyaux sauvages étales d'un corps de nombres. Théorie des nombres. Années 1998/2001. Besançon: Université de Franche-Comté. Publ. Math. UFR Sci. Tech. Besançon (2002; Zbl 1161.11395)].NEWLINENEWLINEFor the entire collection see [Zbl 1235.00045].