  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.
