A series of completions and localisations take place as above until an n-dimensional local field is reached.
This will not suffice for higher dimensional local fields, since one needs to take into account the topology at the residue level too.
Higher local fields can be endowed with appropriate topologies (not uniquely defined) which address this issue.
Instead, there is a finitely additive translation invariant measure defined on the ring of sets generated by closed balls with respect to two-dimensional discrete valuations on the field, and taking values in formal power series R((X)) over reals.
The appropriate replacement for the multiplicative group becomes the nth Milnor K-group, where n is the dimension of the field, which then appears as the domain of a reciprocity map to the Galois group of the maximal abelian extension over the field.
Even better is to work with the quotient of the nth Milnor K-group by its subgroup of elements divisible by every positive integer.
[7] General higher local class field theory was developed by Kazuya Kato[8] and by Ivan Fesenko.
[9][10] Higher local class field theory in positive characteristic was proposed by Aleksei Parshin.