732:. In this case the reciprocity isomorphism of class field theory (or Artin reciprocity map) also admits an explicit description due to the Kronecker–Weber theorem. However, principal constructions of such more detailed theories for small algebraic number fields are not extendable to the general case of algebraic number fields, and different conceptual principles are in use in the general class field theory.
979:
finite Galois extensions as class field theory does in the abelian case. It also does not include an analog of the existence theorem in class field theory: the concept of class fields is absent in the
Langlands correspondence. There are several other nonabelian theories, local and global, which provide alternatives to the Langlands correspondence point of view.
796:
The first two class field theories were very explicit cyclotomic and complex multiplication class field theories. They used additional structures: in the case of the field of rational numbers they use roots of unity, in the case of imaginary quadratic extensions of the field of rational numbers they
735:
The standard method to construct the reciprocity homomorphism is to first construct the local reciprocity isomorphism from the multiplicative group of the completion of a global field to the Galois group of its maximal abelian extension (this is done inside local class field theory) and then prove
978:
Often, the
Langlands correspondence is viewed as a nonabelian class field theory. If and when it is fully established, it would contain a certain theory of nonabelian Galois extensions of global fields. However, the Langlands correspondence does not include as much arithmetical information about
491:
in the case of local fields with finite residue field and the idele class group in the case of global fields. The finite abelian extension corresponding to an open subgroup of finite index is called the class field for that subgroup, which gave the name to the theory.
958:
for number fields, and
Iwasawa theory for number fields use very explicit but narrow class field theory methods or their generalizations. The open question is therefore to use generalizations of general class field theory in these three directions.
633:
373:, and in particular by developing the notion of class formations. Later, Neukirch found a proof of the main statements of global class field theory without using cohomological ideas. His method was explicit and algorithmic.
266:
910:, which became a standard way to learn class field theory for several generations of number theorists. One drawback of the cohomological method is its relative inexplicitness. As the result of local contributions by
72:
who coined the term before
Hilbert's fundamental papers came out. The relevant ideas were developed in the period of several decades, giving rise to a set of conjectures by Hilbert that were subsequently proved by
926:
and also in relation to the work on explicit reciprocity formulas by many mathematicians, a very explicit and cohomology-free presentation of class field theory was established in the 1990s. (See, for example,
755:
which derives class field theory from axioms of class field theory. This derivation is purely topological group theoretical, while to establish the axioms one has to use the ring structure of the ground field.
852:
However, these very explicit theories could not be extended to more general number fields. General class field theory used different concepts and constructions which work over every global field.
759:
There are methods which use cohomology groups, in particular the Brauer group, and there are methods which do not use cohomology groups and are very explicit and fruitful for applications.
903:
in the 1930s to replace ideal classes, essentially clarifying and simplifying the description of abelian extensions of global fields. Most of the central results were proved by 1940.
379:
Explicit class field theory provides an explicit construction of maximal abelian extensions of a number field in various situations. This portion of the theory consists of
711:
677:
403:
986:, which studies algorithms to restore the original object (e.g. a number field or a hyperbolic curve over it) from the knowledge of its full absolute Galois group or
304:
357:
1038:
773:
The origins of class field theory lie in the quadratic reciprocity law proved by Gauss. The generalization took place as a long-term historical project, involving
1184:
Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of
Shinichi Mochizuki, Eur. J. Math., 2015
847:
819:
539:
887:. The first proofs of class field theory used substantial analytic methods. In the 1930s and subsequently saw the increasing use of infinite extensions and
153:
365:, which describes abelian extensions of local fields, and then use it to construct global class field theory. This was first done by Emil Artin and
1495:
1464:
1357:
1335:
1305:
1240:
724:, and the corresponding maximal abelian extension of the rationals is the field generated by all roots of unity. This is known as the
740:
group of the global field is trivial on the image of the multiplicative group of the global field. The latter property is called the
1487:
510:, the multiplicative group of a local field or the idele class group of the global field, with respect to the natural topology on
82:
1049:
1297:
860:
768:
725:
380:
1531:
987:
883:
was known by 1920 and all the main results by about 1930. One of the last classical conjectures to be proved was the
801:
provided another very explicit class field theory for a class of algebraic number fields. In positive characteristic
1228:
915:
884:
778:
745:
460:
366:
1064:
951:
880:
362:
797:
use elliptic curves with complex multiplication and their points of finite order. Much later, the theory of
69:
64:
is credited as one of pioneers of the notion of a class field. However, this notion was already familiar to
43:
1203:
Class field theory, its three main generalisations, and applications, May 2021, EMS Surveys 8(2021) 107-133
868:
479:. In particular, one wishes to establish a one-to-one correspondence between finite abelian extensions of
406:
376:
Inside class field theory one can distinguish special class field theory and general class field theory.
325:
states that the reciprocity map can be used to give a bijection between the set of abelian extensions of
940:
896:
530:
121:
1277:
1253:
1006:
822:
488:
432:
In modern mathematical language, class field theory (CFT) can be formulated as follows. Consider the
1182:
680:
97:
694:
660:
386:
1139:
1054:
1010:
1002:
983:
972:
919:
892:
421:
1473:
1448:
923:
943:. Very explicit class field theory is used in many subareas of algebraic number theory such as
1491:
1460:
1353:
1331:
1301:
1236:
1131:
968:
729:
417:
132:
113:
65:
1201:
274:
1509:
1423:
1399:
1391:
1371:
1345:
1296:, Translations of Mathematical Monographs, vol. 121 (Second ed.), Providence, RI:
1265:
1249:
1156:
1121:
907:
900:
790:
370:
361:
A standard method for developing global class field theory since the 1930s was to construct
332:
47:
1505:
1367:
1315:
1016:
687:
theory which provides more information. For example, the abelianized absolute Galois group
416:
There are three main generalizations of class field theory: higher class field theory, the
1513:
1501:
1456:
1427:
1403:
1375:
1363:
1327:
1311:
1269:
1009:
of finite type over integers and their appropriate localizations and completions. It uses
752:
500:
456:
1434:
1261:
955:
944:
888:
832:
804:
774:
714:
639:
628:{\displaystyle \operatorname {Gal} (L/K)^{\operatorname {ab} }\to C_{K}/N_{L/K}(C_{L})}
1525:
1411:
1143:
911:
856:
826:
61:
17:
1438:
1289:
1109:
876:
864:
798:
782:
718:
642:
of the Galois group of the extension with the quotient of the idele class group of
475:
in terms of open subgroups of finite index in the topological object associated to
441:
74:
55:
1395:
261:{\displaystyle \theta _{L/F}:C_{F}/{N_{L/F}(C_{L})}\to \operatorname {Gal} (L/F),}
1157:
Reciprocity and IUT, talk at RIMS workshop on IUT Summit, July 2016, Ivan
Fesenko
786:
51:
31:
1352:, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press,
1224:
1059:
872:
713:
is (naturally isomorphic to) an infinite product of the group of units of the
647:
78:
1135:
463:, and it is abelian. The central aims of class field theory are: to describe
967:
There are three main generalizations, each of great interest. They are: the
495:
The fundamental result of general class field theory states that the group
993:
Another natural generalization is higher class field theory, divided into
895:
to give a clearer if more abstract formulation of the central result, the
410:
736:
that the product of all such local reciprocity maps when defined on the
1126:
1110:"Class field theory, its three main generalisations, and applications"
27:
Branch of algebraic number theory concerned with abelian extensions
751:
One of the methods to construct the reciprocity homomorphism uses
737:
467:
in terms of certain appropriate topological objects associated to
1005:
and higher global fields. The latter come as function fields of
657:
For some small fields, such as the field of rational numbers
383:, which can be used to construct the abelian extensions of
829:
used Witt duality to get a very easy description of the
789:
on ideals and completions, the theory of cyclotomic and
899:. An important step was the introduction of ideles by
891:'s theory of their Galois groups. This combined with
58:
fields using objects associated to the ground field.
1019:
835:
807:
697:
663:
542:
483:
and their norm groups in this topological object for
389:
335:
277:
156:
329:and the set of closed subgroups of finite index of
120:. This statement was generalized to the so called
1032:
841:
813:
744:and is a far reaching generalization of the Gauss
705:
671:
627:
397:
351:
298:
260:
88:One of the major results is: given a number field
1013:, and appropriate Milnor K-groups generalize the
859:stimulated further development, which led to the
982:Another generalization of class field theory is
906:Later the results were reformulated in terms of
521:. Equivalently, for any finite Galois extension
1095:
517:related to the specific structure of the field
1483:Grundlehren der mathematischen Wissenschaften
1382:Kawada, Yukiyosi (1955), "Class formations",
8:
1481:
1040:used in one-dimensional class field theory.
1324:Class Field Theory: from theory to practice
922:and a local and global reinterpretation by
471:, to describe finite abelian extensions of
46:whose goal is to describe all the abelian
1125:
1024:
1018:
834:
806:
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698:
696:
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391:
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388:
340:
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286:
282:
276:
244:
219:
202:
198:
193:
188:
182:
165:
161:
155:
147:, this law gives a canonical isomorphism
1235:, Redwood City, Calif.: Addison-Wesley,
1076:
849:-part of the reciprocity homomorphism.
1098:, p. 266, Ch. XI by Helmut Hasse.
420:(or 'Langlands correspondences'), and
143:to be any finite abelian extension of
1171:. Charleston, SC: BookSurge, LLC 2006
1083:
1001:. It describes abelian extensions of
963:Generalizations of class field theory
7:
1114:EMS Surveys in Mathematical Sciences
939:Class field theory is used to prove
428:Formulation in contemporary language
409:to construct abelian extensions of
950:Most main achievements toward the
124:; in the idelic language, writing
25:
1294:Local fields and their extensions
975:, and higher class field theory.
487:. This topological object is the
306:denotes the idelic norm map from
112:is canonically isomorphic to the
1414:(1956), "Class formations. II",
999:higher global class field theory
447:. It is of infinite degree over
314:. This isomorphism is named the
1416:J.Fac. Sci.Univ. Tokyo Sect. 1A
995:higher local class field theory
529:, there is an isomorphism (the
499:is naturally isomorphic to the
42:) is the fundamental branch of
1292:; Vostokov, Sergei V. (2002),
1279:History of class field theory.
1050:Non-abelian class field theory
685:very explicit but too specific
681:quadratic imaginary extensions
622:
609:
573:
564:
549:
252:
238:
229:
225:
212:
1:
1396:10.1215/s0012-7094-55-02217-1
1298:American Mathematical Society
879:and many others. The crucial
769:History of class field theory
1108:Fesenko, Ivan (2021-08-31).
728:, originally conjectured by
706:{\displaystyle \mathbb {Q} }
672:{\displaystyle \mathbb {Q} }
650:of the idele class group of
398:{\displaystyle \mathbb {Q} }
1169:Arithmetic duality theorems
1096:Cassels & Fröhlich 1967
988:algebraic fundamental group
947:and Galois modules theory.
455:of A over K is an infinite
1548:
1478:Algebraische Zahlentheorie
1086:, p. 1, Introduction.
766:
98:maximal abelian unramified
1486:. Vol. 322. Berlin:
1065:Langlands correspondences
885:principalisation property
746:quadratic reciprocity law
683:there is a more detailed
461:compact topological group
1350:Local class field theory
952:Langlands correspondence
881:Takagi existence theorem
363:local class field theory
1258:Algebraic Number Theory
954:for number fields, the
855:The famous problems of
726:Kronecker–Weber theorem
381:Kronecker–Weber theorem
299:{\displaystyle N_{L/F}}
44:algebraic number theory
1482:
1322:Gras, Georges (2003).
1200:Fesenko, Ivan (2021),
1181:Fesenko, Ivan (2015),
1034:
843:
815:
785:and Leopold Kronecker/
742:global reciprocity law
707:
673:
629:
407:complex multiplication
399:
353:
352:{\displaystyle C_{F}.}
300:
262:
104:, the Galois group of
1035:
1033:{\displaystyle K_{1}}
941:Artin-Verdier duality
897:Artin reciprocity law
844:
816:
708:
674:
630:
531:Artin reciprocity map
400:
354:
301:
263:
122:Artin reciprocity law
1455:, Berlin, New York:
1017:
833:
805:
695:
661:
646:by the image of the
540:
501:profinite completion
489:multiplicative group
405:, and the theory of
387:
369:using the theory of
333:
275:
154:
83:Chebotarev's theorem
68:and it was actually
18:Abelian number field
1003:higher local fields
869:Philipp Furtwängler
451:; the Galois group
1532:Class field theory
1453:Class Field Theory
1440:Class Field Theory
1410:Kawada, Yukiyosi;
1254:Fröhlich, Albrecht
1233:Class field theory
1055:Anabelian geometry
1030:
1011:algebraic K-theory
984:anabelian geometry
973:anabelian geometry
929:Class Field Theory
920:Michiel Hazewinkel
893:Pontryagin duality
839:
811:
703:
669:
625:
436:abelian extension
422:anabelian geometry
395:
349:
296:
258:
81:(with the help of
36:class field theory
1497:978-3-540-65399-8
1466:978-3-540-15251-4
1359:978-0-19-504030-2
1346:Iwasawa, Kenkichi
1337:978-3-540-44133-5
1307:978-0-8218-3259-2
1242:978-0-201-51011-9
969:Langlands program
842:{\displaystyle p}
814:{\displaystyle p}
791:Kummer extensions
730:Leopold Kronecker
418:Langlands program
323:existence theorem
133:idele class group
114:ideal class group
48:Galois extensions
16:(Redirected from
1539:
1517:
1485:
1474:Neukirch, JĂĽrgen
1469:
1449:Neukirch, JĂĽrgen
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908:group cohomology
901:Claude Chevalley
863:, and proofs by
861:reciprocity laws
848:
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1488:Springer-Verlag
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1457:Springer-Verlag
1447:
1443:(4.03 ed.)
1435:Milne, James S.
1433:
1409:
1381:
1360:
1344:
1338:
1328:Springer-Verlag
1321:
1308:
1290:Fesenko, Ivan B
1288:
1282:
1276:Conrad, Keith,
1275:
1256:, eds. (1967),
1250:Cassels, J.W.S.
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1243:
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1214:
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924:JĂĽrgen Neukirch
831:
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803:
802:
775:quadratic forms
771:
765:
753:class formation
717:taken over all
715:p-adic integers
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658:
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457:profinite group
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316:reciprocity map
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1412:Satake, Ichiro
1407:
1390:(2): 165–177,
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1358:
1342:
1336:
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1262:Academic Press
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1120:(1): 107–133.
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1042:
1027:
1023:
964:
961:
956:BSD conjecture
945:Iwasawa theory
936:
933:
931:by Neukirch.)
889:Wolfgang Krull
838:
810:
767:Main article:
764:
761:
701:
667:
640:abelianization
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619:
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611:
606:
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440:of a local or
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92:, and writing
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1384:Duke Math. J.
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1167:Milne, J. S.
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857:David Hilbert
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850:
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828:
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749:
747:
743:
739:
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719:prime numbers
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690:
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142:
139:, and taking
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111:
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100:extension of
99:
95:
91:
86:
84:
80:
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67:
63:
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45:
41:
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33:
19:
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1225:Artin, Emil
787:Kurt Hensel
781:', work of
777:and their '
32:mathematics
1514:0956.11021
1428:0101.02902
1404:0067.01904
1376:0604.12014
1270:0153.07403
1229:Tate, John
1217:References
1084:Milne 2020
1060:Frobenioid
873:Emil Artin
1144:239667749
1136:2308-2151
1071:Citations
916:John Tate
574:→
547:
411:CM-fields
236:
230:→
159:θ
66:Kronecker
1526:Category
1476:(1999).
1451:(1986),
1437:(2020),
1348:(1986),
1231:(1990),
1044:See also
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1007:schemes
799:Shimura
763:History
679:or its
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