Editing Local class field theory (section)

==Approaches to local class field theory==
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''<sup>&times;</sup>=''K''\{0}. For a finite abelian extension ''L'' of ''K'' the reciprocity map induces an isomorphism of the quotient group ''K''<sup>&times;</sup>/''N''(''L''<sup>&times;</sup>) of ''K''<sup>&times;</sup> by the [[norm group]] ''N''(''L''<sup>&times;</sup>) of the extension ''L''<sup>&times;</sup> to the Galois group Gal(''L''/''K'')
of the extension.<ref name="d1xPt">[[Ivan Fesenko|Fesenko, Ivan]] and Vostokov, Sergei, [http://www.maths.nott.ac.uk/personal/ibf/book/book.html ''Local Fields and their Extensions''], 2nd ed., [[American Mathematical Society]], 2002, {{isbn|0-8218-3259-X}}</ref>

The existence theorem in local class field theory establishes a one-to-one correspondence between open subgroups of finite [[index of a subgroup|index]] in the multiplicative group ''K''<sup>&times;</sup> and finite abelian extensions of the field ''K''. For a finite abelian extension ''L'' of ''K'' the corresponding open subgroup of finite index is the norm group ''N''(''L''<sup>&times;</sup>). The reciprocity map sends higher groups of units to higher ramification subgroups.<ref name="d1xPt" /><sup>Ch.&nbsp;4</sup>

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.<ref name="ksS2P">{{cite journal|title=Sergei V Vostokov, Explicit formulas for the Hilbert symbol, In Invitation to higher local fields |journal=Geometry & Topology Monographs |volume=3 |year=2000 |pages=81–90 |doi=10.2140/gtm.2000.3 |url=http://msp.org/gtm/2000/03/ |editor-last1=Fesenko |editor-last2=Kurihara |editor-first1=Ivan |editor-first2=Masato|url-access=subscription }}</ref>

There are [[cohomology|cohomological]] approaches and non-cohomological approaches to local class field theory. Cohomological approaches tend to be non-explicit, since they use the [[cup product]] of the first [[Galois cohomology]] groups.

For various approaches to local class field theory see Ch. IV and sect. 7 Ch. IV of.<ref name="d1xPt" />  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.