Reduction in K-theory of some infinite extensions of number fields

Abstract

For the cyclotomic extension $F(\mu_{\infty})=\bigcup_{m\geq 1} F(\mu_m)$ of a number field $F,$ we prove that the reduction map $K_{2n{+}1}(F(\mu_{\infty}))\longrightarrow K_{2n{+}1}(\kappa_{{\tilde v}}),$ when restricted to nontorsion elements, is surjective. Here $\kappa_{{\tilde v}}$ denotes the residue field at a prime ${\tilde v}$ of $F(\mu_{\infty}).$