The norm residue symbol for higher local fields

Abstract

In this paper we investigate the Kummer pairing associated to an arbitrary (one-dimensional) formal group. In particular, we obtain formulae describing the values of the pairing in terms of multidimensional p-adic differentiation, the logarithm of the formal group, the generalized trace and the norm on Milnor K-groups. The results are a generalization to higher-dimensional local fields of Kolyvagin's explicit reciprocity laws. In particular, they constitute a generalization of the reciprocity laws of Artin-Hasse, Iwasawa and Wiles.