On finite layers of [formula omitted]-extensions and [formula omitted

Abstract

Let F denote a number field. We study a relation between the subgroup of elements whose lth roots generate extensions of F which are contained in a Zl-extension of F and a certain kernel of Milnor's K-group defined by Tate. We prove that both groups can be described in terms of a norm compatible sequence over the cyclotomic Zl-extension of F.