On finite layers of ZlZl-extensions and K2K2

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 ZlZl-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 ZlZl-extension of F.