38:
103:
692:
225:. In complex analysis, sheets can't simply fold over along a line (one variable), or codimension one subspace in the general case. The ramification set (branch locus on the base, double point set above) will be two real dimensions lower than the ambient
1028:
1741:
527:
598:
1494:
1360:
1298:
1181:
971:
726:
1329:
1267:
1212:
1126:
757:
558:
469:
434:
375:
289:
1806:
1422:
1236:
1150:
890:
781:
340:
1663:
838:
1628:
1394:
940:
1458:
403:
251:
1870:
1850:
1826:
1765:
1687:
1091:
1067:
910:
858:
805:
489:
316:
590:
1941:
155:
976:
229:, and so will not separate it into two 'sides', locally―there will be paths that trace round the branch locus, just as in the example. In
1692:
1553:
1543:
1933:
2018:
31:
1505:
494:
139:
2023:
687:{\displaystyle {\mathfrak {p}}\cdot {\mathcal {O}}_{L}={\mathfrak {p}}_{1}^{e_{1}}\cdots {\mathfrak {p}}_{k}^{e_{k}}}
1995:
2028:
1467:
158:
should multiply by the number of sheets; ramification can therefore be detected by some dropping from that. The
1557:
85:
257:
81:
differing in sign. The term is also used from the opposite perspective (branches coming together) as when a
260:
means a prime ideal factoring in an extension so as to give some repeated prime ideal factors. Namely, let
296:
49:
marked with dots, where the fibers consist of one and two points (marked in black), respectively. The map
1334:
1272:
1155:
945:
700:
1891:
1303:
1241:
1186:
1100:
1070:
731:
532:
443:
408:
349:
263:
37:
1510:
The more detailed analysis of ramification in number fields can be carried out using extensions of the
1770:
1403:
1217:
1131:
871:
762:
321:
1874:
1633:
1588:
1565:
1561:
234:
143:
1527:
1094:
230:
183:
1968:
1919:
810:
1579:. This generalizes the notions in algebraic number theory, local fields, and Dedekind domains.
1937:
1906:
1901:
1601:
1592:
1043:
1955:
1549:
1519:
1039:
437:
292:
222:
115:
1951:
1372:
918:
1959:
1947:
1572:
1461:
131:
107:
1435:
380:
1896:
1855:
1835:
1811:
1750:
1672:
1076:
1052:
895:
843:
790:
474:
301:
74:
563:
2012:
1511:
1429:
102:
1523:
1035:
221:
codimension one, the local complex example sets the pattern for higher-dimensional
97:
82:
343:
206:
70:
17:
2000:
1518:
question. In that case a quantitative measure of ramification is defined for
1031:
210:
166:
mapping shows this as a local pattern: if we exclude 0, looking at 0 < |
88:
at a point of a space, with some collapsing of the fibers of the mapping.
226:
171:
62:
41:
Schematic depiction of ramification: the fibers of almost all points in
1038:. The analogy with the Riemann surface case was already pointed out by
175:
1023:{\displaystyle {\mathcal {O}}_{L}/{\mathfrak {p}}{\mathcal {O}}_{L}}
182:-th power map (Euler–Poincaré characteristic 0), but with the whole
1996:"Splitting and ramification in number fields and Galois extensions"
1534:(non-tame) ramification. This goes beyond the geometric analogue.
101:
36:
1630:
be a morphism of schemes. The support of the quasicoherent sheaf
1736:{\displaystyle f\left(\operatorname {Supp} \Omega _{X/Y}\right)}
1526:
moves field elements with respect to the metric. A sequence of
201:
In geometric terms, ramification is something that happens in
1310:
1248:
1193:
1107:
1009:
983:
738:
615:
539:
508:
450:
415:
356:
270:
237:, by analogy, it also happens in algebraic codimension one.
45:
below consist of three points, except for two points in
1396:
are all relatively prime to the residue characteristic
130: = 0. This is the standard local picture in
1858:
1838:
1814:
1773:
1753:
1695:
1675:
1636:
1604:
1470:
1438:
1406:
1375:
1337:
1306:
1275:
1244:
1220:
1189:
1158:
1134:
1103:
1079:
1055:
979:
948:
921:
898:
874:
846:
813:
793:
765:
734:
703:
601:
566:
560:. This ideal may or may not be prime, but for finite
535:
497:
477:
446:
411:
383:
352:
324:
304:
266:
1852:is also of locally finite presentation we say that
1864:
1844:
1820:
1800:
1759:
1735:
1681:
1657:
1622:
1488:
1452:
1416:
1388:
1354:
1323:
1292:
1261:
1230:
1206:
1175:
1144:
1120:
1085:
1061:
1022:
965:
934:
904:
884:
852:
832:
799:
775:
751:
720:
686:
584:
552:
521:
483:
463:
428:
397:
369:
334:
310:
283:
1970:The Rising Sea: Foundations of algebraic geometry
522:{\displaystyle {\mathfrak {p}}{\mathcal {O}}_{L}}
246:In algebraic extensions of the rational numbers
252:Splitting of prime ideals in Galois extensions
190: – 1 being the 'lost' points as the
1929:Grundlehren der mathematischen Wissenschaften
1432:theory. A finite generically Ă©tale extension
8:
1927:
1530:is defined, reifying (amongst other things)
592:, it has a factorization into prime ideals:
1591:in algebraic geometry. It serves to define
1489:{\displaystyle \operatorname {Tr} :B\to A}
53:is said to be ramified in these points of
1857:
1837:
1813:
1782:
1778:
1772:
1752:
1718:
1714:
1694:
1689:and the image of the ramification locus,
1674:
1645:
1641:
1635:
1603:
1469:
1442:
1437:
1408:
1407:
1405:
1380:
1374:
1346:
1340:
1339:
1336:
1315:
1309:
1308:
1305:
1284:
1278:
1277:
1274:
1253:
1247:
1246:
1243:
1222:
1221:
1219:
1198:
1192:
1191:
1188:
1167:
1161:
1160:
1157:
1136:
1135:
1133:
1112:
1106:
1105:
1102:
1078:
1054:
1014:
1008:
1007:
1000:
999:
994:
988:
982:
981:
978:
957:
951:
950:
947:
926:
920:
897:
876:
875:
873:
845:
818:
812:
792:
767:
766:
764:
743:
737:
736:
733:
712:
706:
705:
702:
676:
671:
666:
660:
659:
647:
642:
637:
631:
630:
620:
614:
613:
603:
602:
600:
565:
544:
538:
537:
534:
513:
507:
506:
499:
498:
496:
476:
455:
449:
448:
445:
420:
414:
413:
410:
387:
382:
361:
355:
354:
351:
326:
325:
323:
303:
275:
269:
268:
265:
27:Branching out of a mathematical structure
1238:is ramified. The latter is an ideal of
186:the Euler–Poincaré characteristic is 1,
126:mapping in the complex plane, near
69:is 'branching out', in the way that the
1587:There is also corresponding notion of
134:theory, of ramification of order
118:, the basic model can be taken as the
1879:
405:we can consider the ring of integers
7:
1269:and is divisible by the prime ideal
1409:
1355:{\displaystyle {\mathfrak {p}}_{i}}
1341:
1293:{\displaystyle {\mathfrak {p}}_{i}}
1279:
1223:
1176:{\displaystyle {\mathfrak {p}}_{i}}
1162:
1137:
1001:
966:{\displaystyle {\mathfrak {p}}_{i}}
952:
877:
768:
721:{\displaystyle {\mathfrak {p}}_{i}}
707:
661:
632:
604:
500:
327:
1775:
1711:
1638:
1522:, basically by asking how far the
1324:{\displaystyle {\mathcal {O}}_{L}}
1262:{\displaystyle {\mathcal {O}}_{L}}
1207:{\displaystyle {\mathcal {O}}_{L}}
1121:{\displaystyle {\mathcal {O}}_{K}}
973:. An equivalent condition is that
752:{\displaystyle {\mathcal {O}}_{L}}
553:{\displaystyle {\mathcal {O}}_{L}}
464:{\displaystyle {\mathcal {O}}_{K}}
429:{\displaystyle {\mathcal {O}}_{L}}
370:{\displaystyle {\mathcal {O}}_{K}}
284:{\displaystyle {\mathcal {O}}_{K}}
142:for the effect of mappings on the
25:
1554:ramification theory of valuations
1544:Ramification theory of valuations
1464:is tame if and only if the trace
1428:. This condition is important in
1967:Vakil, Ravi (18 November 2017).
1034:element: it is not a product of
170:| < 1 say, we have (from the
1801:{\displaystyle \Omega _{X/Y}=0}
1417:{\displaystyle {\mathfrak {p}}}
1231:{\displaystyle {\mathfrak {p}}}
1145:{\displaystyle {\mathfrak {p}}}
1049:The ramification is encoded in
885:{\displaystyle {\mathfrak {p}}}
776:{\displaystyle {\mathfrak {p}}}
335:{\displaystyle {\mathfrak {p}}}
138:. It occurs for example in the
1614:
1480:
1369:when the ramification indices
579:
567:
1:
1658:{\displaystyle \Omega _{X/Y}}
1097:. The former is an ideal of
942:is greater than one for some
728:are distinct prime ideals of
194:sheets come together at
156:Euler–Poincaré characteristic
1506:Ramification of local fields
1046:in the nineteenth century.
2045:
1924:Algebraische Zahlentheorie
1541:
1503:
1152:if and only if some ideal
833:{\displaystyle e_{i}>1}
249:
241:In algebraic number theory
95:
77:, can be seen to have two
29:
1932:. Vol. 322. Berlin:
1623:{\displaystyle f:X\to Y}
377:. For a field extension
178:mapped to itself by the
2019:Algebraic number theory
258:algebraic number theory
140:Riemann–Hurwitz formula
1928:
1866:
1846:
1822:
1802:
1761:
1737:
1683:
1659:
1624:
1490:
1454:
1418:
1390:
1356:
1325:
1294:
1263:
1232:
1208:
1177:
1146:
1122:
1087:
1063:
1024:
967:
936:
906:
886:
854:
834:
801:
777:
753:
722:
688:
586:
554:
523:
485:
465:
430:
399:
371:
336:
312:
297:algebraic number field
285:
154:In a covering map the
111:
58:
1892:Eisenstein polynomial
1867:
1847:
1823:
1803:
1762:
1738:
1684:
1660:
1625:
1583:In algebraic geometry
1491:
1455:
1419:
1391:
1389:{\displaystyle e_{i}}
1357:
1326:
1295:
1264:
1233:
1209:
1178:
1147:
1128:and is divisible by
1123:
1088:
1071:relative discriminant
1064:
1025:
968:
937:
935:{\displaystyle e_{i}}
907:
887:
855:
835:
802:
778:
754:
723:
689:
587:
555:
524:
486:
466:
431:
400:
372:
337:
313:
286:
150:In algebraic topology
105:
40:
1856:
1836:
1812:
1771:
1751:
1693:
1673:
1634:
1602:
1468:
1436:
1404:
1373:
1365:The ramification is
1335:
1304:
1273:
1242:
1218:
1187:
1156:
1132:
1101:
1077:
1053:
977:
946:
919:
896:
872:
844:
811:
791:
763:
732:
701:
599:
564:
533:
495:
475:
444:
409:
381:
350:
322:
302:
264:
30:For other uses, see
1830:formally unramified
1589:unramified morphism
1556:studies the set of
1528:ramification groups
1453:{\displaystyle B/A}
683:
654:
398:{\displaystyle L/K}
217:codimension two is
174:point of view) the
122: →
92:In complex analysis
2024:Algebraic topology
1862:
1842:
1818:
1798:
1757:
1733:
1679:
1667:ramification locus
1655:
1620:
1514:, because it is a
1486:
1450:
1414:
1386:
1352:
1321:
1290:
1259:
1228:
1204:
1173:
1142:
1118:
1095:relative different
1083:
1059:
1020:
963:
932:
914:ramification index
902:
882:
868:. In other words,
860:; otherwise it is
850:
830:
797:
773:
749:
718:
684:
658:
629:
582:
550:
519:
481:
461:
426:
395:
367:
332:
308:
281:
231:algebraic geometry
112:
110:of the square root
59:
1943:978-3-540-65399-8
1907:Branched covering
1902:Puiseux expansion
1865:{\displaystyle f}
1845:{\displaystyle f}
1821:{\displaystyle f}
1760:{\displaystyle f}
1682:{\displaystyle f}
1520:Galois extensions
1086:{\displaystyle L}
1062:{\displaystyle K}
1044:Heinrich M. Weber
905:{\displaystyle L}
853:{\displaystyle i}
800:{\displaystyle L}
491:), and the ideal
484:{\displaystyle L}
311:{\displaystyle K}
223:complex manifolds
198: = 0.
16:(Redirected from
2036:
2029:Complex analysis
2005:
1984:
1982:
1980:
1975:
1963:
1931:
1920:Neukirch, JĂĽrgen
1871:
1869:
1868:
1863:
1851:
1849:
1848:
1843:
1827:
1825:
1824:
1819:
1807:
1805:
1804:
1799:
1791:
1790:
1786:
1766:
1764:
1763:
1758:
1743:, is called the
1742:
1740:
1739:
1734:
1732:
1728:
1727:
1726:
1722:
1688:
1686:
1685:
1680:
1664:
1662:
1661:
1656:
1654:
1653:
1649:
1629:
1627:
1626:
1621:
1550:valuation theory
1495:
1493:
1492:
1487:
1462:Dedekind domains
1459:
1457:
1456:
1451:
1446:
1423:
1421:
1420:
1415:
1413:
1412:
1395:
1393:
1392:
1387:
1385:
1384:
1361:
1359:
1358:
1353:
1351:
1350:
1345:
1344:
1330:
1328:
1327:
1322:
1320:
1319:
1314:
1313:
1299:
1297:
1296:
1291:
1289:
1288:
1283:
1282:
1268:
1266:
1265:
1260:
1258:
1257:
1252:
1251:
1237:
1235:
1234:
1229:
1227:
1226:
1213:
1211:
1210:
1205:
1203:
1202:
1197:
1196:
1182:
1180:
1179:
1174:
1172:
1171:
1166:
1165:
1151:
1149:
1148:
1143:
1141:
1140:
1127:
1125:
1124:
1119:
1117:
1116:
1111:
1110:
1092:
1090:
1089:
1084:
1068:
1066:
1065:
1060:
1040:Richard Dedekind
1029:
1027:
1026:
1021:
1019:
1018:
1013:
1012:
1005:
1004:
998:
993:
992:
987:
986:
972:
970:
969:
964:
962:
961:
956:
955:
941:
939:
938:
933:
931:
930:
911:
909:
908:
903:
891:
889:
888:
883:
881:
880:
866:
865:
859:
857:
856:
851:
839:
837:
836:
831:
823:
822:
806:
804:
803:
798:
782:
780:
779:
774:
772:
771:
758:
756:
755:
750:
748:
747:
742:
741:
727:
725:
724:
719:
717:
716:
711:
710:
693:
691:
690:
685:
682:
681:
680:
670:
665:
664:
653:
652:
651:
641:
636:
635:
625:
624:
619:
618:
608:
607:
591:
589:
588:
585:{\displaystyle }
583:
559:
557:
556:
551:
549:
548:
543:
542:
528:
526:
525:
520:
518:
517:
512:
511:
504:
503:
490:
488:
487:
482:
470:
468:
467:
462:
460:
459:
454:
453:
438:integral closure
435:
433:
432:
427:
425:
424:
419:
418:
404:
402:
401:
396:
391:
376:
374:
373:
368:
366:
365:
360:
359:
341:
339:
338:
333:
331:
330:
317:
315:
314:
309:
293:ring of integers
290:
288:
287:
282:
280:
279:
274:
273:
256:Ramification in
116:complex analysis
21:
2044:
2043:
2039:
2038:
2037:
2035:
2034:
2033:
2009:
2008:
1994:
1991:
1978:
1976:
1973:
1966:
1944:
1934:Springer-Verlag
1918:
1915:
1888:
1854:
1853:
1834:
1833:
1810:
1809:
1774:
1769:
1768:
1749:
1748:
1710:
1703:
1699:
1691:
1690:
1671:
1670:
1637:
1632:
1631:
1600:
1599:
1593:Ă©tale morphisms
1585:
1573:extension field
1546:
1540:
1508:
1502:
1500:In local fields
1496:is surjective.
1466:
1465:
1434:
1433:
1402:
1401:
1376:
1371:
1370:
1338:
1333:
1332:
1331:precisely when
1307:
1302:
1301:
1276:
1271:
1270:
1245:
1240:
1239:
1216:
1215:
1190:
1185:
1184:
1159:
1154:
1153:
1130:
1129:
1104:
1099:
1098:
1075:
1074:
1051:
1050:
1030:has a non-zero
1006:
980:
975:
974:
949:
944:
943:
922:
917:
916:
894:
893:
870:
869:
863:
862:
842:
841:
814:
809:
808:
789:
788:
761:
760:
735:
730:
729:
704:
699:
698:
672:
643:
612:
597:
596:
562:
561:
536:
531:
530:
505:
493:
492:
473:
472:
447:
442:
441:
412:
407:
406:
379:
378:
353:
348:
347:
320:
319:
300:
299:
267:
262:
261:
254:
248:
243:
203:codimension two
152:
132:Riemann surface
108:Riemann surface
100:
94:
75:complex numbers
35:
28:
23:
22:
18:Tamely ramified
15:
12:
11:
5:
2042:
2040:
2032:
2031:
2026:
2021:
2011:
2010:
2007:
2006:
1990:
1989:External links
1987:
1986:
1985:
1964:
1942:
1914:
1911:
1910:
1909:
1904:
1899:
1897:Newton polygon
1894:
1887:
1884:
1861:
1841:
1817:
1797:
1794:
1789:
1785:
1781:
1777:
1756:
1731:
1725:
1721:
1717:
1713:
1709:
1706:
1702:
1698:
1678:
1665:is called the
1652:
1648:
1644:
1640:
1619:
1616:
1613:
1610:
1607:
1584:
1581:
1542:Main article:
1539:
1536:
1512:p-adic numbers
1504:Main article:
1501:
1498:
1485:
1482:
1479:
1476:
1473:
1449:
1445:
1441:
1411:
1383:
1379:
1349:
1343:
1318:
1312:
1287:
1281:
1256:
1250:
1225:
1201:
1195:
1170:
1164:
1139:
1115:
1109:
1082:
1058:
1017:
1011:
1003:
997:
991:
985:
960:
954:
929:
925:
901:
879:
849:
829:
826:
821:
817:
796:
770:
746:
740:
715:
709:
695:
694:
679:
675:
669:
663:
657:
650:
646:
640:
634:
628:
623:
617:
611:
606:
581:
578:
575:
572:
569:
547:
541:
516:
510:
502:
480:
458:
452:
436:(which is the
423:
417:
394:
390:
386:
364:
358:
329:
307:
278:
272:
247:
244:
242:
239:
151:
148:
93:
90:
73:function, for
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2041:
2030:
2027:
2025:
2022:
2020:
2017:
2016:
2014:
2003:
2002:
1997:
1993:
1992:
1988:
1972:
1971:
1965:
1961:
1957:
1953:
1949:
1945:
1939:
1935:
1930:
1925:
1921:
1917:
1916:
1912:
1908:
1905:
1903:
1900:
1898:
1895:
1893:
1890:
1889:
1885:
1883:
1881:
1877:
1876:
1859:
1839:
1831:
1815:
1795:
1792:
1787:
1783:
1779:
1754:
1746:
1729:
1723:
1719:
1715:
1707:
1704:
1700:
1696:
1676:
1668:
1650:
1646:
1642:
1617:
1611:
1608:
1605:
1596:
1594:
1590:
1582:
1580:
1578:
1574:
1570:
1567:
1563:
1559:
1555:
1551:
1545:
1537:
1535:
1533:
1529:
1525:
1521:
1517:
1513:
1507:
1499:
1497:
1483:
1477:
1474:
1471:
1463:
1447:
1443:
1439:
1431:
1430:Galois module
1427:
1399:
1381:
1377:
1368:
1363:
1362:is ramified.
1347:
1316:
1285:
1254:
1199:
1168:
1113:
1096:
1080:
1072:
1056:
1047:
1045:
1041:
1037:
1036:finite fields
1033:
1015:
995:
989:
958:
927:
923:
915:
899:
867:
847:
827:
824:
819:
815:
794:
786:
744:
713:
677:
673:
667:
655:
648:
644:
638:
626:
621:
609:
595:
594:
593:
576:
573:
570:
545:
514:
478:
456:
439:
421:
392:
388:
384:
362:
345:
305:
298:
294:
276:
259:
253:
245:
240:
238:
236:
232:
228:
224:
220:
216:
212:
208:
204:
199:
197:
193:
189:
185:
181:
177:
173:
169:
165:
162:→
161:
157:
149:
147:
145:
141:
137:
133:
129:
125:
121:
117:
109:
104:
99:
91:
89:
87:
84:
80:
76:
72:
68:
64:
56:
52:
48:
44:
39:
33:
19:
1999:
1977:. Retrieved
1969:
1923:
1873:
1829:
1808:we say that
1745:branch locus
1744:
1666:
1597:
1586:
1576:
1568:
1547:
1531:
1524:Galois group
1515:
1509:
1425:
1424:, otherwise
1397:
1366:
1364:
1048:
913:
892:ramifies in
861:
784:
696:
255:
218:
214:
202:
200:
195:
191:
187:
179:
167:
163:
159:
153:
135:
127:
123:
119:
113:
98:Branch point
83:covering map
78:
67:ramification
66:
60:
54:
50:
46:
42:
32:Ramification
783:is said to
344:prime ideal
207:knot theory
86:degenerates
71:square root
2013:Categories
2001:PlanetMath
1960:0956.11021
1913:References
1880:Vakil 2017
1875:unramified
1558:extensions
1538:In algebra
864:unramified
697:where the
250:See also:
106:Using the
96:See also:
1776:Ω
1712:Ω
1708:
1639:Ω
1615:→
1562:valuation
1481:→
1214:dividing
1032:nilpotent
840:for some
656:⋯
610:⋅
233:over any
213:); since
211:monodromy
1922:(1999).
1886:See also
227:manifold
172:homotopy
79:branches
63:geometry
1952:1697859
1832:and if
1093:by the
1073:and in
1069:by the
912:if the
759:. Then
291:be the
219:complex
1979:5 June
1958:
1950:
1940:
1571:to an
1552:, the
785:ramify
318:, and
295:of an
209:, and
205:(like
176:circle
1974:(PDF)
1878:(see
1767:. If
1566:field
1564:of a
1560:of a
1516:local
235:field
144:genus
1981:2019
1938:ISBN
1705:Supp
1598:Let
1532:wild
1426:wild
1367:tame
1042:and
825:>
215:real
184:disk
1956:Zbl
1882:).
1872:is
1828:is
1747:of
1669:of
1575:of
1548:In
1460:of
1400:of
1300:of
1183:of
807:if
787:in
529:of
471:in
440:of
346:of
114:In
61:In
2015::
1998:.
1954:.
1948:MR
1946:.
1936:.
1926:.
1595:.
1472:Tr
342:a
146:.
65:,
2004:.
1983:.
1962:.
1860:f
1840:f
1816:f
1796:0
1793:=
1788:Y
1784:/
1780:X
1755:f
1730:)
1724:Y
1720:/
1716:X
1701:(
1697:f
1677:f
1651:Y
1647:/
1643:X
1618:Y
1612:X
1609::
1606:f
1577:K
1569:K
1484:A
1478:B
1475::
1448:A
1444:/
1440:B
1410:p
1398:p
1382:i
1378:e
1348:i
1342:p
1317:L
1311:O
1286:i
1280:p
1255:L
1249:O
1224:p
1200:L
1194:O
1169:i
1163:p
1138:p
1114:K
1108:O
1081:L
1057:K
1016:L
1010:O
1002:p
996:/
990:L
984:O
959:i
953:p
928:i
924:e
900:L
878:p
848:i
828:1
820:i
816:e
795:L
769:p
745:L
739:O
714:i
708:p
678:k
674:e
668:k
662:p
649:1
645:e
639:1
633:p
627:=
622:L
616:O
605:p
580:]
577:K
574::
571:L
568:[
546:L
540:O
515:L
509:O
501:p
479:L
457:K
451:O
422:L
416:O
393:K
389:/
385:L
363:K
357:O
328:p
306:K
277:K
271:O
196:z
192:n
188:n
180:n
168:z
164:z
160:z
136:n
128:z
124:z
120:z
57:.
55:Y
51:f
47:Y
43:Y
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
