801:, which is a formal product of prime ideals (including, possibly, archimedean ones). The ray class field is the maximal abelian extension unramified outside the primes dividing the modulus and satisfying a particular ramification condition at the primes dividing the modulus. The Hilbert class field is then the ray class field with respect to the trivial modulus
290:
210:
851:
427:
886:
159:
332:
236:
364:
213:
1068:
953:
798:
937:
812:
is the ray class field with respect to the modulus consisting of all infinite primes. For example, the argument above shows that
241:
1056:
974:
429:
has class number 3. Its
Hilbert class field can be formed by adjoining a root of x - x - 1, which has discriminant -23.
1108:
130:
171:
815:
518:
514:
394:
35:
856:
713:
664:
17:
970:
758:
547:
510:
136:
1048:
734:
638:
295:
1092:
790:
550:. The existence of the Hilbert class field is a valuable tool in studying the structure of the
1064:
949:
611:
551:
74:
66:
975:"Allgemeiner Existenzbeweis für den Klassenkörper eines beliebigen algebraischen Zahlkörpers"
1028:
994:
986:
941:
762:
1006:
963:
432:
To see why ramification at the archimedean primes must be taken into account, consider the
1060:
1002:
998:
959:
794:
651:
574:
436:
339:
218:
466:
admits finite abelian extensions of degree greater than 1 in which all finite primes of
754:
349:
101:
1102:
539:
623:
619:
93:
54:
28:
96:(the classical ideal theoretic interpretation) but also at the infinite places of
770:
671:
600:
433:
78:
1088:
1019:
Hilbert, David (1902) , "Über die
Theorie der relativ-Abel'schen Zahlkörper",
945:
38:
1087:
This article incorporates material from
Existence of Hilbert class field on
534:
The existence of a (narrow) Hilbert class field for a given number field
494:
there is ramification at the archimedean places: the real embeddings of
1033:
990:
446:. This field has class number 1 and discriminant 12, but the extension
470:
are unramified. This doesn't contradict the
Hilbert class field of
1044:). See the Introduction chapter of the notes, especially p. 4.
1041:
1040:
J. S. Milne, Class Field Theory (Course notes available at
458:
of discriminant 9=3 is unramified at all prime ideals in
285:{\displaystyle L=\mathbb {Q} ({\sqrt {-3}},{\sqrt {5}})}
346:
has class number 2. Hence, its
Hilbert class field is
859:
818:
737:
satisfying the first, second, and fourth properties.
397:
352:
298:
244:
221:
174:
139:
1053:
Advanced topics in the arithmetic of elliptic curves
498:extend to complex (rather than real) embeddings of
880:
845:
421:
358:
326:
284:
230:
204:
153:
478:itself: every proper finite abelian extension of
1093:Creative Commons Attribution/Share-Alike License
482:must ramify at some place, and in the extension
334:and so is an everywhere unramified extension of
521:at a generator for the ring of integers (as a
442:obtained by adjoining the square root of 3 to
205:{\displaystyle K=\mathbb {Q} ({\sqrt {-15}})}
8:
88:In this context, the Hilbert class field of
1079:Class field theory: From theory to practice
904:
846:{\displaystyle \mathbb {Q} ({\sqrt {3}},i)}
422:{\displaystyle \mathbb {Q} ({\sqrt {-23}})}
1032:
916:
881:{\displaystyle \mathbb {Q} ({\sqrt {3}})}
868:
861:
860:
858:
827:
820:
819:
817:
406:
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182:
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173:
147:
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138:
897:
543:
112:(rather than to a complex embedding of
381:this becomes the principal ideal ((1+
7:
14:
517:is generated by the value of the
65:is canonically isomorphic to the
716:of in the ideal class group of
686:decomposes into the product of
513:, the Hilbert class field of an
781:gives the Hilbert class field.
338:, and it is abelian. Using the
165:is its own Hilbert class field.
108:extends to a real embedding of
1091:, which is licensed under the
875:
865:
840:
824:
566:also satisfies the following:
416:
403:
315:
305:
279:
256:
199:
186:
154:{\displaystyle K=\mathbb {Q} }
92:is not just unramified at the
1:
1057:Graduate Texts in Mathematics
853:is the narrow class field of
327:{\displaystyle 225=(-15)^{2}}
1059:, vol. 151, New York:
1042:http://www.jmilne.org/math/
749:is imaginary quadratic and
366:. A non-principal ideal of
131:unique factorization domain
125:If the ring of integers of
49:equals the class number of
1125:
946:10.1007/978-0-387-72490-4
932:Childress, Nancy (2009),
519:elliptic modular function
515:imaginary quadratic field
797:with respect to a given
562:The Hilbert class field
665:principal ideal theorem
18:algebraic number theory
1077:Gras, Georges (2005),
882:
847:
759:complex multiplication
741:Explicit constructions
654:of the ring extension
511:complex multiplication
423:
360:
328:
286:
232:
206:
155:
979:Mathematische Annalen
883:
848:
769:, then adjoining the
558:Additional properties
424:
361:
329:
287:
233:
207:
156:
1081:, New York: Springer
1049:Silverman, Joseph H.
971:Furtwängler, Philipp
857:
816:
395:
350:
342:, one can show that
296:
242:
219:
172:
137:
45:. Its degree over
573:is a finite Galois
548:Philipp Furtwängler
538:was conjectured by
231:{\displaystyle -15}
133:, in particular if
22:Hilbert class field
1109:Class field theory
1034:10.1007/BF02415486
991:10.1007/BF01448421
934:Class field theory
915:Theorem II.4.1 of
878:
843:
810:narrow class field
793:, one studies the
791:class field theory
554:of a given field.
419:
356:
324:
282:
228:
202:
151:
75:Frobenius elements
1070:978-0-387-94325-1
955:978-0-387-72489-8
873:
832:
612:ideal class group
552:ideal class group
540:David Hilbert
509:By the theory of
414:
359:{\displaystyle L}
292:has discriminant
277:
267:
197:
100:. That is, every
67:ideal class group
1116:
1082:
1073:
1037:
1036:
1021:Acta Mathematica
1015:
1014:
1013:
966:
919:
913:
907:
905:Furtwängler 1906
902:
887:
885:
884:
879:
874:
869:
864:
852:
850:
849:
844:
833:
828:
823:
763:ring of integers
699:prime ideals in
546:) and proved by
428:
426:
425:
420:
415:
407:
402:
387:
386:
376:
375:
365:
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357:
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1124:
1123:
1119:
1118:
1117:
1115:
1114:
1113:
1099:
1098:
1076:
1071:
1061:Springer-Verlag
1047:
1018:
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969:
956:
931:
928:
923:
922:
914:
910:
903:
899:
894:
855:
854:
814:
813:
795:ray class field
787:
785:Generalizations
743:
724:
707:
694:
685:
662:
652:principal ideal
649:
598:
589:
560:
532:
437:quadratic field
393:
392:
384:
382:
373:
371:
348:
347:
340:Minkowski bound
314:
294:
293:
240:
239:
217:
216:
170:
169:
135:
134:
122:
36:maximal abelian
12:
11:
5:
1122:
1120:
1112:
1111:
1101:
1100:
1084:
1083:
1074:
1069:
1045:
1038:
1016:
967:
954:
927:
924:
921:
920:
917:Silverman 1994
908:
896:
895:
893:
890:
877:
872:
867:
863:
842:
839:
836:
831:
826:
822:
786:
783:
755:elliptic curve
742:
739:
733:is the unique
727:
726:
720:
703:
690:
681:
668:
658:
645:
635:
608:
594:
585:
559:
556:
531:
528:
527:
526:
507:
430:
418:
413:
410:
405:
401:
389:
355:
321:
317:
313:
310:
307:
304:
301:
281:
276:
271:
266:
263:
258:
254:
250:
247:
227:
224:
201:
196:
193:
188:
184:
180:
177:
166:
149:
145:
142:
121:
118:
102:real embedding
13:
10:
9:
6:
4:
3:
2:
1121:
1110:
1107:
1106:
1104:
1097:
1096:
1094:
1090:
1080:
1075:
1072:
1066:
1062:
1058:
1054:
1050:
1046:
1043:
1039:
1035:
1030:
1027:(1): 99–131,
1026:
1022:
1017:
1008:
1004:
1000:
996:
992:
988:
984:
980:
976:
972:
968:
965:
961:
957:
951:
947:
943:
939:
935:
930:
929:
925:
918:
912:
909:
906:
901:
898:
891:
889:
870:
837:
834:
829:
811:
806:
804:
800:
796:
792:
784:
782:
780:
776:
772:
768:
764:
760:
756:
752:
748:
740:
738:
736:
732:
723:
719:
715:
711:
706:
702:
698:
693:
689:
684:
680:
676:
673:
669:
666:
661:
657:
653:
650:extends to a
648:
644:
640:
636:
633:
629:
625:
621:
617:
613:
609:
606:
602:
597:
593:
588:
584:
580:
576:
572:
569:
568:
567:
565:
557:
555:
553:
549:
545:
541:
537:
529:
524:
520:
516:
512:
508:
505:
501:
497:
493:
489:
485:
481:
477:
473:
469:
465:
461:
457:
453:
449:
445:
441:
438:
435:
431:
411:
408:
390:
380:
377:)/2), and in
369:
353:
345:
341:
337:
319:
311:
308:
302:
299:
274:
269:
264:
261:
248:
245:
225:
222:
215:
194:
191:
178:
175:
167:
164:
143:
140:
132:
128:
124:
123:
119:
117:
115:
111:
107:
103:
99:
95:
94:finite places
91:
86:
84:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
41:extension of
40:
37:
33:
30:
26:
23:
19:
1086:
1085:
1078:
1052:
1024:
1020:
1010:, retrieved
982:
978:
936:, New York:
933:
911:
900:
809:
807:
802:
788:
778:
774:
766:
750:
746:
744:
730:
728:
721:
717:
709:
704:
700:
696:
691:
687:
682:
678:
674:
659:
655:
646:
642:
631:
627:
624:Galois group
615:
604:
601:class number
595:
591:
586:
582:
578:
570:
563:
561:
535:
533:
522:
503:
499:
495:
491:
487:
483:
479:
475:
471:
467:
463:
459:
455:
451:
447:
443:
439:
378:
367:
343:
335:
238:. The field
214:discriminant
162:
126:
113:
109:
105:
97:
89:
87:
82:
79:prime ideals
70:
62:
58:
55:Galois group
50:
46:
42:
31:
29:number field
24:
21:
15:
985:(1): 1–37,
771:j-invariant
672:prime ideal
1089:PlanetMath
1012:2009-08-21
999:37.0243.02
926:References
620:isomorphic
391:The field
39:unramified
729:In fact,
575:extension
525:-module).
409:−
370:is (2,(1+
309:−
262:−
223:−
192:−
1103:Category
1051:(1994),
973:(1906),
938:Springer
708:, where
590:, where
120:Examples
53:and the
1007:1511392
964:2462595
799:modulus
761:by the
712:is the
622:to the
599:is the
581:and =
542: (
530:History
383:√
372:√
161:, then
34:is the
1067:
1005:
997:
962:
952:
753:is an
670:Every
637:Every
474:being
73:using
20:, the
892:Notes
757:with
735:field
714:order
639:ideal
630:over
462:, so
388:)/2).
129:is a
61:over
27:of a
1065:ISBN
950:ISBN
808:The
610:The
544:1902
434:real
168:Let
77:for
1029:doi
995:JFM
987:doi
942:doi
789:In
777:to
773:of
765:of
745:If
677:of
641:of
626:of
618:is
614:of
603:of
577:of
374:−15
300:225
212:of
116:).
104:of
81:in
69:of
57:of
16:In
1105::
1063:,
1055:,
1025:26
1023:,
1003:MR
1001:,
993:,
983:63
981:,
977:,
960:MR
958:,
948:,
940:,
888:.
805:.
695:/
667:).
506:).
490:)/
454:)/
412:23
312:15
226:15
195:15
85:.
1095:.
1031::
989::
944::
876:)
871:3
866:(
862:Q
841:)
838:i
835:,
830:3
825:(
821:Q
803:1
779:K
775:A
767:K
751:A
747:K
731:E
725:.
722:K
718:O
710:f
705:E
701:O
697:f
692:K
688:h
683:K
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675:P
663:(
660:E
656:O
647:K
643:O
634:.
632:K
628:E
616:K
607:.
605:K
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500:K
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492:K
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484:K
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450:(
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440:K
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404:(
400:Q
385:5
379:L
368:K
354:L
344:K
336:K
320:2
316:)
306:(
303:=
280:)
275:5
270:,
265:3
257:(
253:Q
249:=
246:L
200:)
187:(
183:Q
179:=
176:K
163:K
148:Q
144:=
141:K
127:K
114:E
110:E
106:K
98:K
90:K
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63:K
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