Inmathematics,local class field theory (LCFT), introduced byHelmut Hasse,^[1] is the study ofabelian extensions oflocal fields; here, "local field" means a field which is complete with respect to anabsolute value or adiscrete valuation with a finiteresidue field: hence every local field isisomorphic (as atopological field) to thereal numbersR, thecomplex numbersC, afinite extension of thep-adic numbersQ_p (wherep is anyprime number), or the field offormal Laurent seriesF_q((T)) over afinite fieldF_q.

Local class field theory gives a description of theGalois groupG of the maximal abelian extension of a local fieldK via the reciprocity map which acts from the multiplicative groupK^×=K\{0}. For a finite abelian extensionL ofK the reciprocity map induces an isomorphism of the quotient groupK^×/N(L^×) ofK^× by thenorm groupN(L^×) of the extensionL^× to the Galois group Gal(L/K)of the extension.^[2]

The existence theorem in local class field theory establishes a one-to-one correspondence between open subgroups of finiteindex in the multiplicative groupK^× and finite abelian extensions of the fieldK. For a finite abelian extensionL ofK the corresponding open subgroup of finite index is the norm groupN(L^×). The reciprocity map sends higher groups of units to higher ramification subgroups.^[2]Ch. 4

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.^[3]

There arecohomological approaches and non-cohomological approaches to local class field theory. Cohomological approaches tend to be non-explicit, since they use thecup product of the firstGalois cohomology groups.

For various approaches to local class field theory see Ch. IV and sect. 7 Ch. IV of.^[2] They include the Hasse approach of using theBrauer group, cohomological approaches, the explicit methods ofJürgen Neukirch,Michiel Hazewinkel, theLubin-Tate theory and others.

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.^[3]ch. V

Explicit p-class field theory for local fields withperfect 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 byIvan Fesenko.^[4][5]Fesenko's noncommutative local class field theory for arithmetically profinite Galois extensions of local fields studies appropriate local reciprocity cocycle map and its properties.^[6] This arithmetic theory can be viewed as an alternative to therepresentation-theoreticallocal Langlands correspondence.

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

Higher local class field theory is part ofhigher class field theory which studies abelian extensions (resp. abelian covers) ofrational function fields ofproperregularschemesflat over integers.

• Fesenko, Ivan; Vostokov, Sergey (2002),Local Fields and their Extensions (2nd ed.), American Mathematical Society,ISBN 978-0-19-504030-2
• Fesenko, Ivan B.; Kurihara, Masato, eds. (2000),Invitation to Higher Local Fields, Geometry & Topology Monographs, vol. 3 (First ed.), University of Warwick:Mathematical Sciences Publishers,doi:10.2140/gtm.2000.3,ISSN 1464-8989,Zbl 0954.00026
• Iwasawa, Kenkichi (1986),Local class field theory, Oxford Science Publications, The Clarendon Press Oxford University Press,ISBN 978-0-19-504030-2,MR 0863740
• Neukirch, Jürgen (1986),Class field theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 280, Berlin, New York:Springer-Verlag,ISBN 978-3-540-15251-4,MR 0819231
• Serre, Jean-Pierre (1967), "Local class field theory", in Cassels, John William Scott; Fröhlich, Albrecht (eds.),Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 128–161,ISBN 978-0-9502734-2-6,MR 0220701
• Serre, Jean-Pierre (1979) [1962],Corps Locaux (English translation: Local Fields), Graduate Texts in Mathematics, vol. 67, Berlin, New York:Springer-Verlag,ISBN 978-0-387-90424-5,MR 0150130

• Class field theory

• CS1 German-language sources (de)