Existence theorem for higher local fields

A field K is called an n-dimensional local field if there is a sequence of fields kn, . . . , k0 satisfying the following conditions: k0 is a finite field, ki is a complete discrete valuation field with residue field ki−1 for i = 1, . . . , n, and kn = K . In [9] we defined a canonical homomorphism from the n th Milnor group Kn(K) (cf. [14]) of an n-dimensional local field K to the Galois group Gal(Kab/K) of the maximal abelian extension of K and generalized the familiar results of the usual local class field theory to the case of arbitrary dimension except the “existence theorem”. An essential difficulty with the existence theorem lies in the fact that K (resp. the multiplicative group K ) has no appropriate topology in the case where n > 2 (resp. n > 3 ) which would be compatible with the ring (resp. group) structure and which would take the topologies of the residue fields into account. Thus we abandon the familiar tool “topology” and define the openness of subgroups and the continuity of maps from a new point of view. In the following main theorems the words “open” and “continuous” are not used in the topological sense. They are explained below.