On cyclotomic elements and cyclotomic subgroups in \(K_{2}\) of a field (Q2833582)

For a field \(F\) containing a primitive \(n\)-th root of unity \(\zeta_n\), the Tate-Mekurjev-Suslin theorem states that the \(n\)-torsion subgroup of \(K_2(F)\) is characterized by \(K_2(F)[n]=\{\zeta_n,F^\ast\}\). But the condition \(\zeta_n\in F\) is too restrictive: for example, in \(K_2(\mathbb Q)\), only elements of order 2 can be expressed in this way. For a given \(n\), \textit{J. Browkin} [Lect. Notes Math. 966, 1--6 (1982; Zbl 0502.12009)] introduced the subset \(G_n(F)\) of \(K_2(F)\) consisting of \textit{cyclotomic elements} \(c_n(a):=\{a,\Phi_n(a)\}\), where \(\Phi_n(X)\) is the \(n\)-th cyclotomic polynomial and \(a,\Phi_n(a)\in F^\ast\), and he showed that \(G_n(F)\subset K_2[n]\). Although in some particular cases \(G_n(F)\) is a group, Browkin [loc. cit.] conjectured that for any \(n\neq 1,2,3,6\) and any field \(F,G_n(F)\) is not a group. The best result in this direction is that, for a prime \(l\neq \operatorname{ch}(F)\), \(l\geq 5\) such that \(\Phi_l(X)\) is irreducible, Browkin's conjecture holds for the rational function field \(F(X)\). In the general case, the difficulty of the conjecture is that there seems to exist an ``inner structure'' in \(G_n(F)\), i.e., a partial multiplication structure or a subgroup.NEWLINENEWLINEA subgroup of \(K_2(F)\) will be called \textit{cyclotomic} if it is contained in \(G_n(F)\). In this paper, the authors address the following three problems:NEWLINENEWLINE1) How many non trivial cyclotomic elements are there in a subgroup of \(K_2(F)\) generated by finitely many ``essentially distinct'' (we do not recall the precise meaning) cyclotomic elements in \(G_n(F)\)?NEWLINENEWLINE2) When does \(K_2(F)\) contain a non trivial cyclotomic subgroup?NEWLINENEWLINE3) How many cyclotomic subgroups are there in a subgroup of \(K_2(F)\) generated by finitely many essentially distinct cyclotomic elements in \(G_n(F)\)?NEWLINENEWLINEFor the rational function field \(F(X)\), the authors can determine the exact number of non trivial cyclotomic elements and of non trivial cyclotomic subgroups in a subgroup generated by some kind of cyclotomic elements in \(G_l(F(X))\) where \(l\) is a prime \(\neq \operatorname{ch}(F)\) (the result is too complicated to be restated here). Note that since essential use is made of the existence of a non trivial derivation in \(F(X)\), the proof does not carry over to number fields.NEWLINENEWLINEAs for number fields, hints at the deep ``non closedness'' of the cyclotomic elements in \(K_2(F)\) are provided by the following: given a number field \(F\) and a positive \(n\neq 1,4,8,12\), if there is a prime \(p\) such that \(p^2\) divides \(n\), the authors can construct infinitely many non trivial cyclotomic elements \(\alpha_i\in G_n(F)\) such that \(\langle \alpha^p_1\rangle\subset\langle\alpha^p_1,\alpha^p_2\rangle\subset\dots\) and \(\langle\alpha^p_1,\alpha^p_2,\dots\rangle\cap G_n(F)=(1)\). This implies Browkin's conjecture for integers \(n\neq 1,4,8,12\) which are not square free (a result previously proved by \textit{K. Xu} and \textit{M. Liu} [Sci. China, Ser. A 51, No. 7, 1187--1195 (2008; Zbl 1162.11054)]). In the same vein, the authors conjecture that for a number field \(F\) and a prime \(p>5\), \(K_2(F)\) contains no cyclotomic subgroup of order \(p\).