On cyclotomic elements and cyclotomic subgroups in $K_{2}$ of a field
Volume 175 / 2016
Abstract
The problem of expressing an element of $K_2(F)$ in a more explicit form give rise to many papers. To avoid a restrictive condition in a work of Tate, Browkin considered cyclotomic elements as candidates for elements with an explicit form. In this paper, we modify Browkin’s conjecture about cyclotomic elements into more precise forms, in particular we introduce the concept of cyclotomic subgroups. In the rational function field cases, we completely determine the exact number of cyclotomic elements and of cyclotomic subgroups contained in a subgroup generated by finitely many essentially distinct cyclotomic elements; while in the number field cases, first, some number fields $F$ are constructed so that $K_2(F)$ contains respectively at least one, three and five nontrivial cyclotomic subgroups; then using Faltings’ theorem on the 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.
