670:
381:
997:
136:
564:
386:
where the α run over all the roots of the characteristic polynomial of α other than α itself. The different ideal is generated by the differents of all integers α in
571:
316:
929:
272:
745:
44:
1405:, vol. 77, translated by George U. Brauer; Jay R. Goldman; with the assistance of R. Kotzen, New York–Heidelberg–Berlin:
1491:
1321:
705:. Since the relative discriminant is the norm of the relative different it is the square of a class in the class group of
923:
1577:
1543:
1452:
1414:
1380:
399:
1603:
1483:
1444:
155:
97:
515:
1527:
1402:
1372:
142:
17:
1343:
33:
509:
85:
1501:
1331:
1019:
922:(e) − 1. The differential exponent can be computed from the orders of the
1587:
1553:
1509:
1462:
1424:
1390:
1360:
665:{\displaystyle \delta _{L/K}=\{x\in O_{L}:x\mathrm {d} y=0{\text{ for all }}y\in O_{L}\}.}
8:
1569:
1562:
1433:
1313:
407:
1469:
1573:
1539:
1517:
1487:
1448:
1410:
1376:
1317:
1583:
1549:
1531:
1505:
1458:
1420:
1386:
1364:
1339:
1305:
1015:
212:
52:
29:
1535:
1497:
1440:
1406:
1327:
901:
885:
459:
715:
1597:
445:
376:{\displaystyle \delta (\alpha )=\prod \left({\alpha -\alpha ^{(i)}}\right)\ }
1522:
411:
864:) is greater than 1. The precise exponent to which a ramified prime
1348:
Abhandlungen der Königlichen
Gesellschaft der Wissenschaften zu Göttingen
689:
675:
403:
158:
as quadratic form need not be +1 (in fact this happens only for the case
48:
40:
1026:
is the minimal polynomial for α then the different is generated by
255:
448:
of the relative different is then equal to the relative discriminant Δ
436:
is defined in a similar manner for an extension of number fields
1006:
The different may be defined for an extension of local fields
508:
The relative different equals the annihilator of the relative
28:) is defined to measure the (possible) lack of duality in the
1018:, generated by a primitive element α which also generates a
856:
is ramified, that is, if and only if the ramification index
1371:, Cambridge Studies in Advanced Mathematics, vol. 27,
932:
780:
to a power higher than 1: this occurs if and only if
574:
518:
319:
100:
1439:(2nd, substantially revised and extended ed.),
1435:
Elementary and analytic theory of algebraic numbers
1561:
1432:
991:
664:
558:
375:
130:
992:{\displaystyle \sum _{i=0}^{\infty }(|G_{i}|-1).}
1595:
1014:. In this case we may take the extension to be
1359:
1236:
1234:
1177:
1158:
1156:
1081:
51:of the ring of integers. It was introduced by
1479:Grundlehren der mathematischen Wissenschaften
1092:
1090:
1477:
904:the differential exponent lies in the range
656:
596:
131:{\displaystyle x\mapsto \mathrm {tr} ~x^{2}}
1399:Lectures on the theory of algebraic numbers
1231:
1153:
1141:
395:. This is Dedekind's original definition.
1430:
1344:"Über die Discriminanten endlicher Körper"
1276:
1270:
1252:
1225:
1213:
1111:
1087:
1195:
1107:
1105:
559:{\displaystyle \Omega _{O_{L}/O_{K}}^{1}}
1468:
1338:
1304:
1240:
1173:
1171:
1162:
1147:
1096:
1057:
1046:
473:the relative differents are related by δ
1246:
1219:
1207:
1183:
1129:
1117:
1596:
1310:Elements of the history of mathematics
1102:
1063:
1559:
1516:
1396:
1288:
1264:
1201:
1189:
1168:
1135:
1123:
1069:
417:
1075:
1001:
398:The different is also defined for a
784:divides the relative discriminant Δ
744:The relative different encodes the
310:(and zero otherwise): we may write
13:
949:
714:: indeed, it is the square of the
622:
520:
111:
108:
14:
1615:
1445:PWN-Polish Scientific Publishers
837:divides the relative different δ
238:is the inverse fractional ideal
1282:
1258:
739:
176:Dedekind's complementary module
1530:, vol. 67, translated by
1431:Narkiewicz, Władysław (1990),
1051:
1040:
983:
973:
958:
954:
360:
354:
329:
323:
306:′(α) if α generates the field
104:
1:
1528:Graduate Texts in Mathematics
1403:Graduate Texts in Mathematics
1298:
80:denotes the field trace from
58:
748:data of the field extension
406:. It plays a basic role in
7:
1316:. Berlin: Springer-Verlag.
678:of the relative different δ
72:is the ring of integers of
10:
1620:
1568:(2nd unaltered ed.),
1474:Algebraische Zahlentheorie
1373:Cambridge University Press
1178:Fröhlich & Taylor 1991
1082:Fröhlich & Taylor 1991
924:higher ramification groups
701:, the ring of integers of
688:is always a square in the
1482:. Vol. 322. Berlin:
1356:. Retrieved 5 August 2009
302:is defined to be δ(α) =
267:is equal to the ideal of
1033:
880: − 1 if
868:divides δ is termed the
820:is the factorisation of
768:if the factorisation of
298:with minimal polynomial
194:) is an integer for all
1604:Algebraic number theory
1564:Algebraic Number Theory
1369:Algebraic number theory
926:for Galois extensions:
400:finite degree extension
292:different of an element
143:integral quadratic form
43:. It then encodes the
18:algebraic number theory
1478:
993:
953:
794:. More precisely, if
666:
560:
377:
132:
39:, with respect to the
34:algebraic number field
24:(sometimes simply the
1560:Weiss, Edwin (1976),
1532:Greenberg, Marvin Jay
1397:Hecke, Erich (1981),
994:
933:
870:differential exponent
824:into prime ideals of
667:
561:
378:
228:. By definition, the
133:
86:rational number field
1020:power integral basis
930:
896:. In the case when
776:contains a prime of
572:
516:
317:
242:: it is an ideal of
98:
1279:, pp. 194, 270
637: for all
555:
510:Kähler differential
1570:Chelsea Publishing
1518:Serre, Jean-Pierre
1361:Fröhlich, Albrecht
1204:, pp. 234–236
1150:, pp. 197–198
989:
662:
556:
519:
424:relative different
418:Relative different
408:Pontryagin duality
373:
273:field discriminant
128:
1493:978-3-540-65399-8
1340:Dedekind, Richard
1323:978-3-540-64767-6
1306:Bourbaki, Nicolas
1002:Local computation
756:. A prime ideal
638:
372:
271:generated by the
168:inverse different
117:
1611:
1590:
1567:
1556:
1513:
1481:
1470:Neukirch, Jürgen
1465:
1438:
1427:
1393:
1355:
1335:
1312:. Translated by
1292:
1286:
1280:
1274:
1268:
1262:
1256:
1250:
1244:
1238:
1229:
1223:
1217:
1211:
1205:
1199:
1193:
1187:
1181:
1175:
1166:
1160:
1151:
1145:
1139:
1133:
1127:
1121:
1115:
1109:
1100:
1094:
1085:
1079:
1073:
1067:
1061:
1055:
1049:
1044:
998:
996:
995:
990:
976:
971:
970:
961:
952:
947:
892:does not divide
888:: that is, when
876:and is equal to
671:
669:
668:
663:
655:
654:
639:
636:
625:
614:
613:
592:
591:
587:
565:
563:
562:
557:
554:
549:
548:
547:
538:
533:
532:
382:
380:
379:
374:
370:
369:
365:
364:
363:
213:fractional ideal
137:
135:
134:
129:
127:
126:
115:
114:
53:Richard Dedekind
30:ring of integers
1619:
1618:
1614:
1613:
1612:
1610:
1609:
1608:
1594:
1593:
1580:
1546:
1536:Springer-Verlag
1494:
1484:Springer-Verlag
1455:
1441:Springer-Verlag
1417:
1407:Springer-Verlag
1383:
1324:
1301:
1296:
1295:
1287:
1283:
1277:Narkiewicz 1990
1275:
1271:
1263:
1259:
1253:Narkiewicz 1990
1251:
1247:
1239:
1232:
1226:Narkiewicz 1990
1224:
1220:
1214:Narkiewicz 1990
1212:
1208:
1200:
1196:
1188:
1184:
1176:
1169:
1161:
1154:
1146:
1142:
1134:
1130:
1122:
1118:
1112:Narkiewicz 1990
1110:
1103:
1095:
1088:
1080:
1076:
1068:
1064:
1056:
1052:
1045:
1041:
1036:
1004:
972:
966:
962:
957:
948:
937:
931:
928:
927:
921:
902:wildly ramified
886:tamely ramified
855:
847:if and only if
846:
836:
816:
807:
793:
742:
735:
726:
713:
700:
687:
650:
646:
635:
621:
609:
605:
583:
579:
575:
573:
570:
569:
550:
543:
539:
534:
528:
524:
523:
517:
514:
513:
504:
492:
482:
460:tower of fields
457:
435:
420:
394:
353:
349:
342:
338:
318:
315:
314:
282:
266:
250:
237:
230:different ideal
227:
206:
153:
122:
118:
107:
99:
96:
95:
71:
61:
22:different ideal
12:
11:
5:
1617:
1607:
1606:
1592:
1591:
1578:
1557:
1544:
1514:
1492:
1466:
1453:
1428:
1415:
1394:
1381:
1365:Taylor, Martin
1357:
1336:
1322:
1300:
1297:
1294:
1293:
1281:
1269:
1257:
1245:
1243:, pp. 199
1230:
1218:
1206:
1194:
1182:
1167:
1152:
1140:
1128:
1116:
1101:
1086:
1074:
1062:
1050:
1038:
1037:
1035:
1032:
1003:
1000:
988:
985:
982:
979:
975:
969:
965:
960:
956:
951:
946:
943:
940:
936:
917:
851:
838:
832:
818:
817:
812:
805:
785:
741:
738:
731:
722:
716:Steinitz class
709:
696:
679:
661:
658:
653:
649:
645:
642:
634:
631:
628:
624:
620:
617:
612:
608:
604:
601:
598:
595:
590:
586:
582:
578:
553:
546:
542:
537:
531:
527:
522:
496:
484:
474:
449:
427:
419:
416:
390:
384:
383:
368:
362:
359:
356:
352:
348:
345:
341:
337:
334:
331:
328:
325:
322:
278:
262:
246:
233:
223:
202:
166:). Define the
149:
139:
138:
125:
121:
113:
110:
106:
103:
67:
60:
57:
9:
6:
4:
3:
2:
1616:
1605:
1602:
1601:
1599:
1589:
1585:
1581:
1579:0-8284-0293-0
1575:
1571:
1566:
1565:
1558:
1555:
1551:
1547:
1545:0-387-90424-7
1541:
1537:
1533:
1529:
1525:
1524:
1519:
1515:
1511:
1507:
1503:
1499:
1495:
1489:
1485:
1480:
1475:
1471:
1467:
1464:
1460:
1456:
1454:3-540-51250-0
1450:
1446:
1442:
1437:
1436:
1429:
1426:
1422:
1418:
1416:3-540-90595-2
1412:
1408:
1404:
1400:
1395:
1392:
1388:
1384:
1382:0-521-36664-X
1378:
1374:
1370:
1366:
1362:
1358:
1353:
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1319:
1315:
1314:Meldrum, John
1311:
1307:
1303:
1302:
1290:
1285:
1278:
1273:
1266:
1261:
1255:, p. 166
1254:
1249:
1242:
1241:Neukirch 1999
1237:
1235:
1228:, p. 401
1227:
1222:
1216:, p. 304
1215:
1210:
1203:
1198:
1191:
1186:
1180:, p. 126
1179:
1174:
1172:
1165:, p. 201
1164:
1163:Neukirch 1999
1159:
1157:
1149:
1148:Neukirch 1999
1144:
1138:, p. 121
1137:
1132:
1126:, p. 116
1125:
1120:
1114:, p. 160
1113:
1108:
1106:
1099:, p. 195
1098:
1097:Neukirch 1999
1093:
1091:
1084:, p. 125
1083:
1078:
1071:
1066:
1059:
1058:Bourbaki 1994
1054:
1048:
1047:Dedekind 1882
1043:
1039:
1031:
1029:
1025:
1021:
1017:
1013:
1010: /
1009:
999:
986:
980:
977:
967:
963:
944:
941:
938:
934:
925:
920:
915:
912: +
911:
907:
903:
899:
895:
891:
887:
883:
879:
875:
871:
867:
863:
859:
854:
850:
845:
842: /
841:
835:
831:
827:
823:
815:
811:
804:
800:
797:
796:
795:
792:
789: /
788:
783:
779:
775:
771:
767:
763:
759:
755:
752: /
751:
747:
737:
734:
730:
725:
721:
717:
712:
708:
704:
699:
695:
691:
686:
683: /
682:
677:
672:
659:
651:
647:
643:
640:
632:
629:
626:
618:
615:
610:
606:
602:
599:
593:
588:
584:
580:
576:
567:
551:
544:
540:
535:
529:
525:
511:
506:
503:
500: /
499:
495:
491:
488: /
487:
481:
478: /
477:
472:
469: /
468:
465: /
464:
461:
456:
453: /
452:
447:
446:relative norm
443:
440: /
439:
434:
431: /
430:
425:
415:
413:
412:p-adic fields
409:
405:
401:
396:
393:
389:
366:
357:
350:
346:
343:
339:
335:
332:
326:
320:
313:
312:
311:
309:
305:
301:
297:
293:
288:
286:
281:
277:
274:
270:
265:
261:
257:
252:
249:
245:
241:
236:
231:
226:
222:
218:
214:
210:
205:
201:
197:
193:
190:such that tr(
189:
185:
181:
177:
173:
169:
165:
161:
157:
152:
148:
144:
123:
119:
101:
94:
93:
92:
90:
87:
83:
79:
75:
70:
66:
56:
54:
50:
46:
42:
38:
35:
31:
27:
23:
19:
1563:
1523:Local Fields
1521:
1473:
1434:
1398:
1368:
1351:
1347:
1309:
1284:
1272:
1260:
1248:
1221:
1209:
1197:
1192:, p. 59
1185:
1143:
1131:
1119:
1077:
1072:, p. 50
1065:
1053:
1042:
1027:
1023:
1011:
1007:
1005:
918:
913:
909:
905:
897:
893:
889:
881:
877:
873:
869:
865:
861:
857:
852:
848:
843:
839:
833:
829:
825:
821:
819:
813:
809:
802:
798:
790:
786:
781:
777:
773:
769:
765:
764:ramifies in
761:
757:
753:
749:
746:ramification
743:
740:Ramification
732:
728:
723:
719:
710:
706:
702:
697:
693:
684:
680:
673:
568:
507:
501:
497:
493:
489:
485:
479:
475:
470:
466:
462:
454:
450:
441:
437:
432:
428:
423:
421:
404:local fields
397:
391:
387:
385:
307:
303:
299:
295:
291:
289:
284:
279:
275:
268:
263:
259:
253:
247:
243:
239:
234:
229:
224:
220:
216:
208:
203:
199:
195:
191:
187:
183:
179:
175:
171:
167:
163:
159:
156:discriminant
150:
146:
140:
88:
81:
77:
73:
68:
64:
62:
49:prime ideals
45:ramification
36:
25:
21:
15:
690:class group
676:ideal class
219:containing
178:as the set
172:codifferent
41:field trace
1588:0348.12101
1554:0423.12016
1510:0956.11021
1463:0717.11045
1425:0504.12001
1391:0744.11001
1299:References
1289:Weiss 1976
1265:Weiss 1976
1202:Hecke 1981
1190:Serre 1979
1136:Hecke 1981
1124:Hecke 1981
1070:Serre 1979
256:ideal norm
59:Definition
1354:(2): 1–56
978:−
950:∞
935:∑
736:-module.
644:∈
603:∈
577:δ
521:Ω
351:α
347:−
344:α
336:∏
327:α
321:δ
105:↦
55:in 1882.
47:data for
26:different
1598:Category
1520:(1979),
1472:(1999).
1367:(1991),
1342:(1882),
1308:(1994).
1291:, p. 115
1267:, p. 114
1060:, p. 102
458:. In a
283:of
1502:1697859
1332:1290116
512:module
444:. The
207:, then
91:, then
84:to the
1586:
1576:
1552:
1542:
1508:
1500:
1490:
1461:
1451:
1423:
1413:
1389:
1379:
1330:
1320:
1022:. If
1016:simple
371:
154:. Its
141:is an
116:
76:, and
32:of an
20:, the
1034:Notes
1030:(α).
828:then
727:as a
294:α of
211:is a
1574:ISBN
1540:ISBN
1488:ISBN
1449:ISBN
1411:ISBN
1377:ISBN
1318:ISBN
808:...
718:for
674:The
422:The
410:for
290:The
254:The
1584:Zbl
1550:Zbl
1506:Zbl
1459:Zbl
1421:Zbl
1387:Zbl
908:to
900:is
884:is
872:of
772:in
760:of
692:of
483:= δ
402:of
258:of
215:of
198:in
182:of
174:or
170:or
145:on
63:If
16:In
1600::
1582:,
1572:,
1548:,
1538:,
1534:,
1526:,
1504:.
1498:MR
1496:.
1486:.
1476:.
1457:,
1447:,
1443:;
1419:,
1409:,
1401:,
1385:,
1375:,
1363:;
1352:29
1350:,
1346:,
1328:MR
1326:.
1233:^
1170:^
1155:^
1104:^
1089:^
1028:f'
801:=
566::
505:.
414:.
287:.
251:.
192:xy
186:∈
162:=
78:tr
1512:.
1334:.
1024:f
1012:K
1008:L
987:.
984:)
981:1
974:|
968:i
964:G
959:|
955:(
945:0
942:=
939:i
919:P
916:ν
914:e
910:e
906:e
898:P
894:e
890:P
882:P
878:e
874:P
866:P
862:i
860:(
858:e
853:i
849:P
844:K
840:L
834:i
830:P
826:L
822:p
814:k
810:P
806:1
803:P
799:p
791:K
787:L
782:p
778:L
774:L
770:p
766:L
762:K
758:p
754:K
750:L
733:K
729:O
724:L
720:O
711:K
707:O
703:L
698:L
694:O
685:K
681:L
660:.
657:}
652:L
648:O
641:y
633:0
630:=
627:y
623:d
619:x
616::
611:L
607:O
600:x
597:{
594:=
589:K
585:/
581:L
552:1
545:K
541:O
536:/
530:L
526:O
502:F
498:K
494:δ
490:K
486:L
480:F
476:L
471:F
467:K
463:L
455:K
451:L
442:K
438:L
433:K
429:L
426:δ
392:K
388:O
367:)
361:)
358:i
355:(
340:(
333:=
330:)
324:(
308:K
304:f
300:f
296:K
285:K
280:K
276:D
269:Z
264:K
260:δ
248:K
244:O
240:I
235:K
232:δ
225:K
221:O
217:K
209:I
204:K
200:O
196:y
188:K
184:x
180:I
164:Q
160:K
151:K
147:O
124:2
120:x
112:r
109:t
102:x
89:Q
82:K
74:K
69:K
65:O
37:K
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.