ON cyclotomic elements and cyclotomic subgroups in K_{2} of a field

Abstract: The problem of expressing an element of K_2(F) in a more explicit form gives rise to many works. To avoid a restrictive condition in a work of Tate, Browkin considered cyclotomic elements as the candidate for the element with an explicit form. In this paper, we modify and change Browkin's conjecture about cyclotomic elements into more precise forms, in particular we introduce the conception of cyclotomic subgroup. In the rational function field cases, we determine completely the exact numbers of cyclotomic elements and cyclotomic subgroups contained in a subgroup generated by finitely many different cyclotomic elements, while in the number field cases, using Faltings' theorem on Mordell conjecture we prove that there exist subgroups generated by an infinite number of cyclotomic elements to the power of some prime, which contain no nontrivial cyclotomic elements.