Local class field theory gives a description of the Galois group G of the maximal abelian extension of a local field K via the reciprocity map which acts from the multiplicative group K×=K\{0}.
[3] Using the local reciprocity map, one defines the Hilbert symbol and its generalizations.
Finding explicit formulas for it is one of subdirections of the theory of local fields, it has a long and rich history, see e.g. Sergei Vostokov's review.
IV of [5] They include the Hasse approach of using the Brauer group, cohomological approaches, the explicit methods of Jürgen Neukirch, Michiel Hazewinkel, the Lubin-Tate theory and others.
[9] This arithmetic theory can be viewed as an alternative to the representation theoretical local Langlands correspondence.
Unlike the classical case, Milnor K-groups do not satisfy Galois module descent if
General higher-dimensional local class field theory was developed by K. Kato and I. Fesenko.
abelian covers) of rational function fields of proper regular schemes flat over integers.