Editing Local class field theory (section)

==Generalizations of local class field theory==
Generalizations of local class field theory to local fields with quasi-finite residue field were easy extensions of the theory, obtained by G. Whaples in the 1950s.<ref name="ksS2P" /><sup>ch.&nbsp;V</sup>

Explicit p-class field theory for local fields with [[perfect field|perfect]] and imperfect residue fields which are not finite has to deal with the new issue of norm groups of infinite index. Appropriate theories were constructed by [[Ivan Fesenko]].<ref>{{cite journal |title=Local class field theory: perfect residue field case |author=I. Fesenko | publisher=Russian Academy of Sciences |journal=Izvestiya: Mathematics |volume=43 |number=1 |year=1994 |pages=65–81|doi=10.1070/IM1994v043n01ABEH001559 |bibcode=1994IzMat..43...65F}}</ref><ref>{{Cite journal |last=Fesenko |first=I. |title=On general local reciprocity maps|journal=[[Journal für die reine und angewandte Mathematik]] |volume=473 |year=1996 |pages=207–222}}</ref>
Fesenko's noncommutative local class field theory for arithmetically profinite Galois extensions of local fields studies appropriate local reciprocity cocycle map and its properties.<ref>{{Cite book |last=Fesenko |first=I. |chapter=Nonabelian local reciprocity maps |title=Class Field Theory – Its Centenary and Prospect, Advanced Studies in Pure Math |year=2001 |pages=63–78 |publisher=Mathematical Society of Japan |isbn = 4-931469-11-6}}</ref> This arithmetic theory can be viewed as an alternative to the [[representation theory|representation-theoretical]] [[local Langlands correspondence]].