- Local class field theory
In

mathematics ,**local class field theory**is the study innumber theory of theabelian extension s oflocal field s. It is in itself a rather successful theory, leading to definite conclusions. It is also important for (and was developed to help elucidate) the proofs ofclass field theory itself.The basic theory concerns for a local field "K", the description of the

Galois group "G" of the "maximal" abelian extension of "K". This is closely related to "K"^{×}, themultiplicative group of "K"{0}. These groups cannot be equal: The topological group "G" ispro-finite and socompact . On the other hand "K"^{×}is not compact.Taking the case where "K" is a

finite extension of thep-adic number s**Q**_{p}, we can say more precisely that "K"^{×}has the structure of acartesian product of a compact group with aninfinite cyclic group. The main topological operation is to replace the infinite cyclic group by a group**Z**^{^}, i.e. its pro-finite completion with respect to subgroups of finite index. This can be done by indicating a topology on "K"^{×}, for which we can complete. This, roughly speaking, is then the correct group to identify with "G".The actual isomorphism used is important in practice, and is described in the theory of the

norm residue symbol .For a description of the general case of local class field theory see

class formation .**See also***

Class field theory

*Class formation

*Quasi-finite field **References*** Milne, James, [

*http://www.jmilne.org/math/CourseNotes/math776.html*] .**Class Field Theory**

* Fesenko, Ivan and Vostokov, Sergei, [*http://www.maths.nott.ac.uk/personal/ibf/book/book.html*] , 2nd ed.,**Local Fields and Their Extension**American Mathematical Society , 2002, ISBN 0-8218-3259-X

* Iwasawa, Kenkichi, "Local Class Field Theory",Oxford University Press , 1986, ISBN 0195040309.

* Neukirch, Jürgen, "Class field theory",Springer-Verlag , 1986, ISBN 3-540-15251-2. Chap.III.

* Serre, Jean-Pierre,**Local Class Field Theory**in "Algebraic Number Theory", Proceedings (edd Cassels, J.W.S. and Fröhlich, A),Academic Press , 1967, ISBN 012268950X. Pp. 128-161.

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