The norm residue symbol for higher local fields
  Since the development of higher local class field theory, several explicit
reciprocity laws have been constructed. In particular, there are formulas
describing the higher-dimensional Hilbert symbol given, among others, by M.
Kurihara, A. Zinoviev and S. Vostokov. K. Kato also has explicit formulas for
the higher-dimensional Kummer pairing associated to certain (one-dimensional)
$p$-divisible groups.
In this paper we construct an explicit reciprocity law describing the Kummer
pairing associated to any (one-dimensional) formal group. The formulas are a
generalization to higher-dimensional local fields of Kolyvagin's reciprocity
laws. The formulas obtained describe 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.
In the second part of this paper, we will apply the results obtained here to
give explicit formulas for the generalized Hilbert symbol and the Kummer
pairing associated to a Lubin-Tate formal group. The results obtained in the
second paper constitute a generalization to higher local fields, of the
formulas of Artin-Hasse, K. Iwasawa and A. Wiles.
