The Hilbert symbol in multi-dimensional local fields for Lubin–Tate formal groups

In this paper an explicit formula for the Hilbert pairing between the Milnor $K$-group of a multi-dimensional local field and the multi-dimensional Lubin–Tate formal module is derived. This formula is a generalization of such a formula in one-dimensional case. Here we consider the case, where the penultimate residue field is of characteristic zero.