View source for Local class field theory

In [[mathematics]], '''local class field theory''', introduced by [[Helmut Hasse]],<ref>{{Citation | last1=Hasse | first1=H. | author1-link=Helmut Hasse | title=Die Normenresttheorie relativ-Abelscher Zahlkörper als Klassenkörpertheorie im Kleinen. | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002171171 | language=de | doi=10.1515/crll.1930.162.145 | jfm=56.0165.03 | year=1930 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=1930 | issue=162 | pages=145–154| s2cid=116860448| url-access=subscription }}</ref> is the study of [[abelian extension]]s of [[local field]]s; here, "local field" means a field which is complete with respect to an [[Absolute value (algebra)|absolute value]] or a [[discrete valuation]] with a finite [[residue field]]: hence every local field is [[isomorphic]] (as a [[topological field]]) to the [[real numbers]] '''R''', the [[complex numbers]] '''C''', a [[finite extension]] of the [[p-adic number|''p''-adic number]]s '''Q'''<sub>''p''</sub> (where ''p'' is any [[prime number]]), or the field of [[formal Laurent series]] '''F'''<sub>''q''</sub>((''T'')) over a [[finite field]] '''F'''<sub>''q''</sub>.

==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.

==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]].

==Higher local class field theory==
For a [[Higher local field|higher-dimensional local field]] <math>K</math> there is a higher local reciprocity map which describes abelian extensions of the field in terms of open subgroups of finite index in the [[Milnor K-group]] of the field. Namely, if <math>K</math> is an <math>n</math>-dimensional local field then one uses <math>\mathrm{K}^{\mathrm{M}}_n(K)</math> or its separated quotient endowed with a suitable topology. When <math>n=1</math> the theory becomes the usual local class field theory. Unlike the classical case, Milnor K-groups do not satisfy Galois module descent if <math>n>1</math>. General higher-dimensional local class field theory was developed by [[Kazuya Kato|K. Kato]] and [[Ivan Fesenko|I. Fesenko]].

Higher local class field theory is part of [[higher class field theory]] which studies abelian extensions (resp. abelian covers) of [[Function field (scheme theory)|rational function fields]] of [[proper scheme|proper]] [[Regular scheme|regular]] [[scheme (mathematics)|schemes]] [[flat morphism|flat]] over integers.

==References==
{{reflist}}

==Further reading==
*{{Citation | last1=Fesenko | first1=Ivan | last2=Vostokov | first2=Sergey |title=Local Fields and their Extensions |edition=2nd
 | url=https://www.maths.nottingham.ac.uk/personal/ibf/book/book.html | publisher=American Mathematical Society | isbn=978-0-19-504030-2 | year=2002}}
* {{Citation| editor-last=Fesenko| editor-first=Ivan B.| editor-link=Ivan Fesenko| editor2-last=Kurihara
| editor2-first=Masato| title=Invitation to Higher Local Fields| publisher=[[Mathematical Sciences Publishers]]| location= University of Warwick| year=2000| series=Geometry & Topology Monographs | volume=3| edition=First| doi=10.2140/gtm.2000.3 | issn=1464-8989 | zbl=0954.00026 }}
*{{Citation | last1=Iwasawa | first1=Kenkichi | title=Local class field theory | url=https://books.google.com/books?id=iJ7vAAAAMAAJ | publisher=The Clarendon Press Oxford University Press | series=Oxford Science Publications | isbn=978-0-19-504030-2 | mr=863740 | year=1986}}
*{{Citation | last1=Neukirch | first1=Jürgen | author1-link=Jürgen Neukirch | title=Class field theory | url=https://books.google.com/books?id=5_vuAAAAMAAJ | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-3-540-15251-4 | mr=819231 | year=1986 | volume=280}}
*{{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | editor1-last=Cassels | editor1-first=John William Scott | editor2-last=Fröhlich | editor2-first=Albrecht | title=Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) | url=https://books.google.com/books?id=DQP_RAAACAAJ | publisher=Thompson, Washington, D.C. | isbn=978-0-9502734-2-6 | mr=0220701 | year=1967 | chapter=Local class field theory | pages=128–161}}
*{{Citation | last1=Serre | first1=Jean-Pierre | title=Corps Locaux (English translation: Local Fields) | orig-year=1962 | url=https://books.google.com/books?id=DAxlMdw_QloC | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-90424-5 | mr=0150130 | year=1979 | volume=67}}

[[Category:Class field theory]]