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

Kejian Xu ¹   ; Chaochao Sun ²

¹ School of Mathematics and Statistics Qingdao University Qingdao 266071, China and Institute of Applied Mathematics of Shandong Qingdao University Qingdao 266071, China
² Department of Mathematics Linyi University Linyi 276005, China and School of Mathematics Jilin University Changchun 130012, China

Acta Arithmetica, Tome 175 (2016) no. 1, pp. 1-55

Résumé

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.

Détail

DOI : 10.4064/aa8084-4-2016

Keywords: problem expressing element explicit form rise many papers avoid restrictive condition work tate browkin considered cyclotomic elements candidates elements explicit form paper modify browkin conjecture about cyclotomic elements precise forms particular introduce concept cyclotomic subgroups rational function field cases completely determine exact number cyclotomic elements cyclotomic subgroups contained subgroup generated finitely many essentially distinct cyclotomic elements while number field cases first number fields constructed contains respectively least three five nontrivial cyclotomic subgroups using faltings theorem mordell conjecture prove there exist subgroups generated infinite number cyclotomic elements power prime which contain nontrivial cyclotomic elements