591:
asks for generalizations of the
Kronecker–Weber theorem to base fields other than the rational numbers, and asks for the analogues of the roots of unity for those fields. A different approach to abelian extensions is given by
252:
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1064:, Trudy Mat. Inst. Steklov. (in Russian), vol. 38, Moscow: Izdat. Akad. Nauk SSSR, pp. 382–387,
447:
1077:
878:
530:
442:. Precisely, the Kronecker–Weber theorem states: every finite abelian extension of the rational numbers
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32:
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43:
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537:) published a proof, but this had some gaps and errors that were pointed out and corrected by
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28:
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565:) proved the local Kronecker–Weber theorem which states that any abelian extension of a
502:
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708:
684:
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459:
455:
113:
109:
927:
1010:
Rosen, Michael (1981), "An elementary proof of the local
Kronecker-Weber theorem",
713:"Ein neuer Beweis des Kronecker'schen Fundamentalsatzes über Abel'sche Zahlkörper."
105:
39:
566:
529:) though his argument was not complete for extensions of degree a power of 2.
247:{\displaystyle {\sqrt {5}}=e^{2\pi i/5}-e^{4\pi i/5}-e^{6\pi i/5}+e^{8\pi i/5},}
970:"Two proofs of the Kronecker-Weber theorem "according to Kronecker, and Weber""
629:
Greenberg, M. J. (1974). "An
Elementary Proof of the Kronecker-Weber Theorem".
987:
1135:
1035:
995:
953:
899:
849:
809:
693:
607:
16:
Every finite abelian extension of Q is contained within some cyclotomic field
1059:
477:
cyclotomic field that contains it. The theorem allows one to define the
1126:
1043:
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907:
857:
662:
650:
1082:
Matériaux pour l'histoire des mathématiques au XX siècle (Nice, 1996)
801:
1026:
891:
840:
642:
792:
1078:"On the history of Hilbert's twelfth problem: a comedy of errors"
100:
is contained within some cyclotomic field. In other words, every
776:
96:
provides a partial converse: every finite abelian extension of
824:
Lubin, Jonathan (1981), "The local
Kronecker-Weber theorem",
458:, the field is a subfield of a field obtained by adjoining a
717:
Nachrichten der
Gesellschaft der Wissenschaften zu Göttingen
613:, in Adhikari, S. D.; Katre, S. A.; Thakur, Dinesh (eds.),
446:
is a subfield of a cyclotomic field. That is, whenever an
876:(1965), "Formal complex multiplication in local fields",
1110:
Weber, H. (1886), "Theorie der Abel'schen Zahlkörper",
617:, Bhaskaracharya Pratishthana, Pune, pp. 135–146,
330:{\displaystyle {\sqrt {-3}}=e^{2\pi i/3}-e^{4\pi i/3},}
434:
The
Kronecker–Weber theorem can be stated in terms of
84:{\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times }}
408:{\displaystyle {\sqrt {3}}=e^{\pi i/6}-e^{5\pi i/6}.}
343:
259:
125:
49:
928:"Formal moduli for one-parameter formal Lie groups"
569:can be constructed using cyclotomic extensions and
407:
329:
246:
83:
1013:Transactions of the American Mathematical Society
827:Transactions of the American Mathematical Society
615:Cyclotomic fields and related topics (Pune, 1999)
975:Journal für die reine und angewandte Mathematik
729:"Über die algebraisch auflösbaren Gleichungen"
932:Bulletin de la Société Mathématique de France
8:
1061:A new proof of the Kronecker-Weber theorem
779:Journal de théorie des nombres de Bordeaux
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116:with rational coefficients. For example,
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53:
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541:. The first complete proof was given by
546:
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493:lies inside the field generated by the
1084:, Sémin. Congr., vol. 3, Paris:
582:
578:
534:
7:
497:-th roots of unity. For example the
1167:Theorems in algebraic number theory
663:"Local class field theory is easy"
14:
521:The theorem was first stated by
1086:Société Mathématique de France
465:For a given abelian extension
72:
50:
1:
1076:Schappacher, Norbert (1998),
631:American Mathematical Monthly
608:"The Kronecker-Weber theorem"
557:Lubin and Tate (
112:can be expressed as a sum of
23:, it can be shown that every
685:10.1016/0001-8708(75)90156-5
755:"Über Abelsche Gleichungen"
430:Field-theoretic formulation
418:The theorem is named after
1183:
771:, Collected works volume 4
745:, Collected works volume 4
1058:Šafarevič, I. R. (1951),
988:10.1515/crll.1981.323.105
589:Hilbert's twelfth problem
462:to the rational numbers.
509:, a fact generalised in
485:as the smallest integer
450:has a Galois group over
671:Advances in Mathematics
94:Kronecker–Weber theorem
21:algebraic number theory
968:Neumann, Olaf (1981),
761:(in German): 845–851,
735:(in German): 365–374,
606:Ghate, Eknath (2000),
501:have as conductor the
448:algebraic number field
409:
331:
248:
85:
879:Annals of Mathematics
759:Berlin K. Akad. Wiss.
733:Berlin K. Akad. Wiss.
585:) gave other proofs.
573:. Hazewinkel (
571:Lubin–Tate extensions
424:Heinrich Martin Weber
410:
332:
249:
86:
33:rational number field
1088:, pp. 243–273,
341:
257:
123:
47:
659:Hazewinkel, Michiel
1157:Class field theory
1127:10.1007/BF02417089
945:10.24033/bsmf.1633
751:Kronecker, Leopold
725:Kronecker, Leopold
719:(in German): 29–39
594:class field theory
581:) and Lubin (
511:class field theory
405:
327:
244:
81:
1162:Cyclotomic fields
1095:978-2-85629-065-1
922:Lubin, Jonathan;
882:, Second Series,
872:Lubin, Jonathan;
420:Leopold Kronecker
349:
268:
131:
102:algebraic integer
29:abelian extension
1174:
1138:
1129:
1113:Acta Mathematica
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802:10.5802/jtnb.507
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628:
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1143:External links
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1020:(2): 599–605,
1007:
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886:(2): 380–387,
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834:(1): 133–138,
821:
786:(2): 555–558,
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709:Hilbert, David
705:
678:(2): 148–181,
655:
637:(6): 601–607.
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507:discriminant
494:
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474:
473:there is a
470:
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451:
443:
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417:
106:Galois group
97:
93:
42:of the form
40:Galois group
35:
18:
1120:: 193–263,
567:local field
454:that is an
1151:Categories
924:Tate, John
874:Tate, John
600:References
489:such that
1136:0001-5962
1036:0002-9947
996:0075-4102
954:0037-9484
938:: 49–59,
900:0003-486X
850:0002-9947
810:1246-7405
793:1108.5671
694:0001-8708
523:Kronecker
505:of their
479:conductor
387:π
376:−
360:π
309:π
298:−
282:π
263:−
226:π
199:π
188:−
172:π
161:−
145:π
77:×
38:, having
926:(1966),
753:(1877),
727:(1853),
711:(1896),
661:(1975),
1104:1640262
1070:0049233
1052:0610968
1044:1999753
1004:0611446
962:0238854
916:0172878
908:1970622
866:0621978
858:1998574
818:2211307
702:0389858
651:2319208
623:1802379
545: (
543:Hilbert
533: (
525: (
517:History
475:minimal
110:abelian
31:of the
1134:
1102:
1092:
1068:
1050:
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1002:
994:
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848:
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739:
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649:
621:
436:fields
104:whose
92:. The
27:is an
1040:JSTOR
904:JSTOR
854:JSTOR
788:arXiv
666:(PDF)
647:JSTOR
611:(PDF)
531:Weber
1132:ISSN
1090:ISBN
1032:ISSN
992:ISSN
950:ISSN
896:ISSN
846:ISSN
806:ISSN
763:ISBN
737:ISBN
690:ISSN
583:1981
579:1981
575:1975
563:1966
559:1965
547:1896
535:1886
527:1853
438:and
422:and
337:and
1122:doi
1022:doi
1018:265
984:doi
980:323
940:doi
888:doi
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680:doi
639:doi
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481:of
469:of
108:is
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1100:MR
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1080:,
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1038:,
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1016:,
1000:MR
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990:,
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958:MR
956:,
948:,
936:94
934:,
930:,
912:MR
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902:,
894:,
884:81
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