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In [[mathematics]], '''local class field theory''', introduced by {{harvs|txt|last=Hasse|year=1930|authorlink=Helmut Hasse}}, is the study  of  [[abelian extension]]s of [[local field]]s; here, "local field'' means : a field which is complete with respect to a discrete valuation, with a finite residue field. It is the analogue for local fields of  global [[class field theory]].  
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Local class field theory gives a description of the [[Galois group]] ''G'' of the maximal abelian extension of  a local field ''K''
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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.  
 
The absolute Galois group ''G'' of ''K'' is compact and the group ''K''<sup>&times;</sup>
is not compact. Taking the case where ''K'' is a [[finite extension]] of the [[p-adic number]]s '''Q'''<sub>p</sub> or formal power series over a finite field, the group ''K''<sup>&times;</sup> is the product of a compact group with an [[infinite cyclic]] group '''Z'''. The main topological operation is to replace ''K''<sup>&times;</sup> by its [[profinite completion]], which is roughly the same as replacing the factor '''Z''' by its profinite completion '''Z'''<sup>^</sup>. The profinite completion of ''K''<sup>&times;</sup> is the group isomorphic  with ''G'' via the local reciprocity map.  
 
The actual isomorphism used and the existence theorem is described in the theory of the [[norm residue symbol]]. There are several different approaches to the theory, using central division algebras or Tate cohomology or an explicit description of the reciprocity map. There are also two different normalizations of the reciprocity map: in the case of an unramified extension, one of them asks that the (arithmetic) Frobenius element corresponds to the elements of  "K" of valuation 1; the other one is the opposite.  
 
 
== Higher local class field theory ==
For a [[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>. Higher dimensional class field theory was pioneered by A.N. Parshin in positive characteristic and [[Kazuya Kato|K. Kato]], [[Ivan Fesenko|I. Fesenko]], Sh. Saito in the general case.
 
== See also ==
 
*[[Lubin–Tate theory]]
* [[Quasi-finite field]]
*[[Local Langlands conjectures]]
*[[Norm group]]
 
==References==
 
* {{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 and Topology Monographs
| volume=3| edition=First| doi=10.2140/gtm.2000.3 | issn=1464-8989 | zbl=0954.00026 }}
* [[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
*{{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=German | 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=162 | pages=145–154}}
*{{Citation | last1=Iwasawa | first1=Kenkichi | title=Local class field theory | url=http://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}}
* Milne, James, [http://www.jmilne.org/math/CourseNotes/cft.html '''Class Field Theory'''].
*{{Citation | last1=Neukirch | first1=Jürgen | author1-link=Jürgen Neukirch | title=Class field theory | url=http://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=http://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 | author1-link=Jean-Pierre Serre | title=Corps Locaux (English translation: Local Fields) | origyear=1962 | url=http://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]]

Latest revision as of 18:40, 25 November 2014

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