1173:
710:
420:
555:
250:
1025:
900:
170:
118:
70:
580:
936:
This name is for several different closely related symbols, such as the Artin symbol or the
Hilbert symbol or Hasse's norm residue symbol. The Hasse norm residue symbol
838:
859:"Chevalley symbol" has several slightly different meanings. It is sometimes used for the Artin symbol for ideles. A variation of this is the Chevalley symbol
321:
459:
1205:
31:
The symbols below are arranged roughly in order of the date they were introduced, which is usually (but not always) in order of increasing generality.
192:
1145:
939:
1095:
A sort of generalization of the
Steinberg symbol to higher algebraic K-theory. It takes a Milnor K-group to an étale cohomology group.
1137:
1181:
862:
1210:
180:
an integer, and takes values 0, 1, or −1. An extension of the Jacobi and
Legendre symbols to more general values of
140:
88:
40:
318:
roots of 1. The power residue symbol can be written in terms of the
Hilbert symbol. The global Hilbert symbol
1105:
128:
an integer, and takes values 0, 1, or −1. An extension of the
Legendre symbol to more general values of
705:{\displaystyle \psi _{L/K}(\alpha )=(\alpha ,L/K)=\left({\frac {L/K}{\alpha }}\right)=((L/K)/\alpha )}
187:
1191:
incorrectly led you here, you may wish to change the link to point directly to the intended article.
933:
1123:
1141:
841:
284:= 2 and the global field is the rationals this is more or less the same as the Jacobi symbol.
1159:
1050:
764:
759:
745:
135:
1155:
1163:
1151:
1110:
35:
25:
930:. The value of the symbol is then the value of the character χ on the usual Artin symbol.
415:{\displaystyle (a,b)_{p}=\left({\frac {a,b}{p}}\right)=\left({\frac {a,b}{p}}\right)_{m}}
550:{\displaystyle \theta _{L/K}(\alpha )=(\alpha ,L/K)=\left({\frac {L/K}{\alpha }}\right)}
287:
1199:
1092:
83:
17:
453:
1039:. It is essentially the same as the local Artin symbol for the localization of
755:, where the roots of unity can be identified with elements of the Galois group.
712:
is defined for α in a ray class group or idele (class) group of a global field
1184:
includes a list of related items that share the same name (or similar names).
1047:. The Hilbert symbol is a special case of it in the case of Kummer extensions.
28:. This article describes the relations between these various generalizations.
1061:). This is a generalization of the local Hilbert symbol to arbitrary fields
1172:
245:{\displaystyle \left({\frac {a}{b}}\right)=\left({\frac {a}{b}}\right)_{m}}
1020:{\displaystyle ((\alpha ,L/K)/p)=\left({\frac {\alpha ,L/K}{p}}\right)}
569:, and takes values in the abelianization of the Galois group Gal(
732:. When α is in the idele group the symbol is sometimes called a
442:, and is equal to the local Hilbert symbol in the completion of
1188:
942:
865:
767:
583:
462:
324:
195:
143:
91:
43:
1019:
894:
832:
704:
549:
414:
244:
164:
112:
64:
895:{\displaystyle \left({\frac {a,\chi }{p}}\right)}
744:can be written in terms of the Artin symbol for
716:, and takes values in the abelianization of Gal(
24:is any of many different generalizations of the
1178:Index of articles associated with the same name
918:, and χ a homomorphism of the Galois group of
456:The local Artin symbol or norm residue symbol
1133:Grundlehren der mathematischen Wissenschaften
8:
1131:
165:{\displaystyle \left({\frac {a}{b}}\right)}
113:{\displaystyle \left({\frac {a}{b}}\right)}
65:{\displaystyle \left({\frac {a}{p}}\right)}
999:
987:
969:
958:
941:
870:
864:
812:
806:
788:
777:
766:
691:
680:
650:
644:
626:
592:
588:
582:
529:
523:
505:
471:
467:
461:
406:
384:
354:
341:
323:
236:
222:
200:
194:
148:
142:
96:
90:
80:an integer, and takes values 0, 1, or −1.
48:
42:
1085:) takes values in the second K-group of
561:a finite extension of the local field
7:
256:in some global field containing the
306:in some local field containing the
272:built from prime ideals coprime to
14:
1206:Set index articles on mathematics
276:. The symbol takes values in the
1171:
977:
966:
946:
943:
796:
785:
771:
768:
740:. The local Hilbert symbol of
699:
688:
674:
671:
634:
614:
608:
602:
513:
493:
487:
481:
438:a finite or infinite place of
338:
325:
1:
577:). The global Artin symbol
1227:
1170:
1128:Algebraische Zahlentheorie
314:) and takes values in the
290:The local Hilbert symbol (
1136:. Vol. 322. Berlin:
260:th roots of 1 ( for some
848:of the Galois extension
728:an abelian extension of
124:a positive odd integer,
1132:
1021:
896:
834:
833:{\displaystyle =\left}
738:Artin–Chevalley symbol
706:
551:
416:
268:a fractional ideal of
246:
166:
114:
66:
1106:Contou-Carrère symbol
1022:
897:
835:
707:
552:
430:in some global field
417:
310:roots of 1 (for some
247:
167:
115:
67:
1035:and α an element of
940:
863:
765:
581:
460:
322:
193:
188:Power residue symbol
141:
89:
41:
934:Norm residue symbol
840:is the same as the
298:) = is defined for
1211:Class field theory
1077:, and the symbol (
1017:
892:
830:
702:
565:, α an element of
547:
412:
242:
162:
110:
62:
1182:set index article
1147:978-3-540-65399-8
1011:
906:a prime ideal of
886:
842:Frobenius element
824:
746:Kummer extensions
662:
541:
400:
370:
280:roots of 1. When
230:
208:
156:
104:
56:
1218:
1192:
1175:
1167:
1135:
1124:Neukirch, Jürgen
1073:are elements of
1051:Steinberg symbol
1026:
1024:
1023:
1018:
1016:
1012:
1007:
1003:
988:
973:
962:
901:
899:
898:
893:
891:
887:
882:
871:
839:
837:
836:
831:
829:
825:
820:
816:
807:
792:
781:
760:Frobenius symbol
734:Chevalley symbol
711:
709:
708:
703:
695:
684:
667:
663:
658:
654:
645:
630:
601:
600:
596:
556:
554:
553:
548:
546:
542:
537:
533:
524:
509:
480:
479:
475:
421:
419:
418:
413:
411:
410:
405:
401:
396:
385:
375:
371:
366:
355:
346:
345:
251:
249:
248:
243:
241:
240:
235:
231:
223:
213:
209:
201:
171:
169:
168:
163:
161:
157:
149:
136:Kronecker symbol
119:
117:
116:
111:
109:
105:
97:
71:
69:
68:
63:
61:
57:
49:
1226:
1225:
1221:
1220:
1219:
1217:
1216:
1215:
1196:
1195:
1194:
1193:
1186:
1185:
1179:
1148:
1138:Springer-Verlag
1122:
1119:
1111:Mennicke symbol
1102:
989:
983:
938:
937:
872:
866:
861:
860:
808:
802:
763:
762:
646:
640:
584:
579:
578:
557:is defined for
525:
519:
463:
458:
457:
422:is defined for
386:
380:
379:
356:
350:
337:
320:
319:
252:is defined for
218:
217:
196:
191:
190:
144:
139:
138:
92:
87:
86:
44:
39:
38:
36:Legendre symbol
26:Legendre symbol
12:
11:
5:
1224:
1222:
1214:
1213:
1208:
1198:
1197:
1177:
1176:
1169:
1168:
1146:
1118:
1115:
1114:
1113:
1108:
1101:
1098:
1097:
1096:
1090:
1065:. The numbers
1048:
1031:is a place of
1027:is defined if
1015:
1010:
1006:
1002:
998:
995:
992:
986:
982:
979:
976:
972:
968:
965:
961:
957:
954:
951:
948:
945:
931:
914:an element of
890:
885:
881:
878:
875:
869:
857:
828:
823:
819:
815:
811:
805:
801:
798:
795:
791:
787:
784:
780:
776:
773:
770:
756:
701:
698:
694:
690:
687:
683:
679:
676:
673:
670:
666:
661:
657:
653:
649:
643:
639:
636:
633:
629:
625:
622:
619:
616:
613:
610:
607:
604:
599:
595:
591:
587:
545:
540:
536:
532:
528:
522:
518:
515:
512:
508:
504:
501:
498:
495:
492:
489:
486:
483:
478:
474:
470:
466:
451:
409:
404:
399:
395:
392:
389:
383:
378:
374:
369:
365:
362:
359:
353:
349:
344:
340:
336:
333:
330:
327:
288:Hilbert symbol
285:
239:
234:
229:
226:
221:
216:
212:
207:
204:
199:
185:
160:
155:
152:
147:
133:
108:
103:
100:
95:
81:
60:
55:
52:
47:
13:
10:
9:
6:
4:
3:
2:
1223:
1212:
1209:
1207:
1204:
1203:
1201:
1190:
1189:internal link
1183:
1174:
1165:
1161:
1157:
1153:
1149:
1143:
1139:
1134:
1129:
1125:
1121:
1120:
1116:
1112:
1109:
1107:
1104:
1103:
1099:
1094:
1093:Galois symbol
1091:
1088:
1084:
1080:
1076:
1072:
1068:
1064:
1060:
1056:
1052:
1049:
1046:
1042:
1038:
1034:
1030:
1013:
1008:
1004:
1000:
996:
993:
990:
984:
980:
974:
970:
963:
959:
955:
952:
949:
935:
932:
929:
925:
921:
917:
913:
909:
905:
888:
883:
879:
876:
873:
867:
858:
855:
851:
847:
844:of the prime
843:
826:
821:
817:
813:
809:
803:
799:
793:
789:
782:
778:
774:
761:
757:
754:
750:
747:
743:
739:
735:
731:
727:
723:
719:
715:
696:
692:
685:
681:
677:
668:
664:
659:
655:
651:
647:
641:
637:
631:
627:
623:
620:
617:
611:
605:
597:
593:
589:
585:
576:
572:
568:
564:
560:
543:
538:
534:
530:
526:
520:
516:
510:
506:
502:
499:
496:
490:
484:
476:
472:
468:
464:
455:
452:
449:
446:at the place
445:
441:
437:
433:
429:
425:
407:
402:
397:
393:
390:
387:
381:
376:
372:
367:
363:
360:
357:
351:
347:
342:
334:
331:
328:
317:
313:
309:
305:
301:
297:
293:
289:
286:
283:
279:
275:
271:
267:
263:
259:
255:
237:
232:
227:
224:
219:
214:
210:
205:
202:
197:
189:
186:
183:
179:
176:any integer,
175:
158:
153:
150:
145:
137:
134:
131:
127:
123:
106:
101:
98:
93:
85:
84:Jacobi symbol
82:
79:
75:
58:
53:
50:
45:
37:
34:
33:
32:
29:
27:
23:
19:
18:number theory
1127:
1086:
1082:
1078:
1074:
1070:
1066:
1062:
1058:
1054:
1044:
1040:
1036:
1032:
1028:
927:
923:
919:
915:
911:
907:
903:
853:
849:
845:
752:
748:
741:
737:
733:
729:
725:
721:
717:
713:
574:
570:
566:
562:
558:
454:Artin symbol
447:
443:
439:
435:
431:
427:
423:
315:
311:
307:
303:
299:
295:
291:
281:
277:
273:
269:
265:
261:
257:
253:
181:
177:
173:
172:defined for
129:
125:
121:
120:defined for
77:
73:
72:defined for
30:
21:
15:
1200:Categories
1164:0956.11021
1117:References
991:α
950:α
880:χ
697:α
660:α
618:α
606:α
586:ψ
539:α
497:α
485:α
465:θ
76:a prime,
1126:(1999).
1100:See also
1156:1697859
1187:If an
1162:
1154:
1144:
724:) for
434:, for
22:symbol
1180:This
1142:ISBN
1069:and
902:for
758:The
426:and
302:and
20:, a
1160:Zbl
1043:at
922:to
852:of
736:or
264:),
16:In
1202::
1158:.
1152:MR
1150:.
1140:.
1130:.
910:,
1166:.
1089:.
1087:F
1083:b
1081:,
1079:a
1075:F
1071:b
1067:a
1063:F
1059:b
1057:,
1055:a
1053:(
1045:p
1041:K
1037:K
1033:K
1029:p
1014:)
1009:p
1005:K
1001:/
997:L
994:,
985:(
981:=
978:)
975:p
971:/
967:)
964:K
960:/
956:L
953:,
947:(
944:(
928:Z
926:/
924:R
920:K
916:K
912:a
908:K
904:p
889:)
884:p
877:,
874:a
868:(
856:.
854:K
850:L
846:P
827:]
822:P
818:K
814:/
810:L
804:[
800:=
797:]
794:P
790:/
786:)
783:K
779:/
775:L
772:(
769:[
753:K
751:/
749:L
742:K
730:K
726:L
722:K
720:/
718:L
714:K
700:)
693:/
689:)
686:K
682:/
678:L
675:(
672:(
669:=
665:)
656:K
652:/
648:L
642:(
638:=
635:)
632:K
628:/
624:L
621:,
615:(
612:=
609:)
603:(
598:K
594:/
590:L
575:K
573:/
571:L
567:K
563:K
559:L
544:)
535:K
531:/
527:L
521:(
517:=
514:)
511:K
507:/
503:L
500:,
494:(
491:=
488:)
482:(
477:K
473:/
469:L
450:.
448:p
444:K
440:K
436:p
432:K
428:b
424:a
408:m
403:)
398:p
394:b
391:,
388:a
382:(
377:=
373:)
368:p
364:b
361:,
358:a
352:(
348:=
343:p
339:)
335:b
332:,
329:a
326:(
316:m
312:m
308:m
304:b
300:a
296:b
294:,
292:a
282:m
278:m
274:m
270:K
266:b
262:m
258:m
254:a
238:m
233:)
228:b
225:a
220:(
215:=
211:)
206:b
203:a
198:(
184:.
182:b
178:a
174:b
159:)
154:b
151:a
146:(
132:.
130:b
126:a
122:b
107:)
102:b
99:a
94:(
78:a
74:p
59:)
54:p
51:a
46:(
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