1288:
951:
357:
1044:
1750:
1701:
1652:
1603:
1554:
1505:
695:
1365:
1121:
825:
756:
106:
475:
1451:
1423:
612:
1139:
251:
The existence, and especially the construction, of a generic polynomial for a given Galois group provides a complete solution to the
843:
1778:
287:
148:
969:
542:
33:
1707:
1658:
1609:
1560:
1511:
1462:
630:
1783:
252:
1306:
1062:
769:
478:
255:
for that group. However, not all Galois groups have generic polynomials, a counterexample being the
134:
713:
63:
446:
409:
The cyclic group construction leads to other classes of generic polynomials; in particular the
491:
1436:
1395:
584:
441:
429:
141:
395:
269:
183:
248:
polynomial, which is generic relative to the rational numbers is called simply generic.
549:
410:
402:
is divisible by eight, and G. W. Smith explicitly constructs such a polynomial in case
216:
can be obtained as the splitting field of a polynomial which is the specialization of
1772:
111:
481:
391:
256:
194:
127:
29:
17:
25:
1425:, is defined as the minimal number of parameters in a generic polynomial for
1372:
Generic polynomials are known for all transitive groups of degree 5 or less.
115:
1283:{\displaystyle x^{5}+(t-3)x^{4}+(s-t+3)x^{3}+(t^{2}-t-2s-1)x^{2}+sx+t}
398:
showed that a cyclic group does not have a generic polynomial if
54:
are indeterminates, the generic polynomial of degree two in
946:{\displaystyle x^{4}-2s(t^{2}+1)x^{2}+s^{2}t^{2}(t^{2}+1)}
517:, including in particular groups of the root systems for
1710:
1661:
1612:
1563:
1514:
1465:
1439:
1398:
1309:
1142:
1065:
972:
846:
772:
716:
633:
587:
552:
of two groups both of which have generic polynomials.
545:
of two groups both of which have generic polynomials.
449:
290:
66:
1761:
Jensen, Christian U., Ledet, Arne, and Yui, Noriko,
1744:
1695:
1646:
1597:
1548:
1499:
1445:
1417:
1359:
1282:
1115:
1038:
945:
819:
750:
689:
606:
469:
351:
100:
352:{\displaystyle x^{n}+t_{1}x^{n-1}+\cdots +t_{n}}
122:has a different, although related, meaning: a
8:
560:
1039:{\displaystyle x^{4}-2stx^{2}+s^{2}t(t-1)}
1730:
1720:
1719:
1718:
1709:
1681:
1671:
1670:
1669:
1660:
1632:
1622:
1621:
1620:
1611:
1583:
1573:
1572:
1571:
1562:
1534:
1524:
1523:
1522:
1513:
1485:
1475:
1474:
1473:
1464:
1438:
1406:
1397:
1330:
1314:
1308:
1259:
1225:
1209:
1175:
1147:
1141:
1086:
1070:
1064:
1012:
999:
977:
971:
928:
915:
905:
892:
873:
851:
845:
802:
780:
771:
721:
715:
654:
638:
632:
592:
586:
459:
454:
448:
343:
318:
308:
295:
289:
74:
65:
1745:{\displaystyle gd_{\mathbb {Q} }S_{5}=2}
1696:{\displaystyle gd_{\mathbb {Q} }D_{5}=2}
1647:{\displaystyle gd_{\mathbb {Q} }S_{4}=2}
1598:{\displaystyle gd_{\mathbb {Q} }D_{4}=2}
1549:{\displaystyle gd_{\mathbb {Q} }S_{3}=1}
1500:{\displaystyle gd_{\mathbb {Q} }A_{3}=1}
421:has a generic polynomial if and only if
690:{\displaystyle x^{3}-tx^{2}+(t-3)x+1}
7:
1360:{\displaystyle x^{5}+sx^{3}-t(x+1)}
1116:{\displaystyle x^{4}+sx^{2}-t(x+1)}
1765:, Cambridge University Press, 2002
1440:
820:{\displaystyle (x^{2}-s)(x^{2}-t)}
14:
1453:if no generic polynomial exists.
204:, and such that every extension
118:, and in this article, the term
557:Examples of generic polynomials
513:Reflection groups defined over
263:Groups with generic polynomials
1354:
1342:
1252:
1218:
1202:
1184:
1168:
1156:
1110:
1098:
1033:
1021:
940:
921:
885:
866:
814:
795:
792:
773:
745:
733:
675:
663:
1:
751:{\displaystyle x^{3}-t(x+1)}
363:is a generic polynomial for
101:{\displaystyle ax^{2}+bx+c.}
232:. This is sometimes called
220:resulting from setting the
149:field of rational functions
1800:
425:is not divisible by eight.
406:is not divisible by eight.
470:{\displaystyle H_{p^{3}}}
236:or relative to the field
147:with coefficients in the
1446:{\displaystyle \infty }
1418:{\displaystyle gd_{F}G}
607:{\displaystyle x^{2}-t}
1746:
1697:
1648:
1599:
1550:
1501:
1447:
1419:
1361:
1284:
1117:
1040:
947:
821:
752:
691:
608:
503:The alternating group
471:
353:
253:inverse Galois problem
102:
1747:
1698:
1649:
1600:
1551:
1502:
1448:
1420:
1362:
1285:
1118:
1041:
948:
822:
753:
692:
609:
548:Any group which is a
541:Any group which is a
472:
354:
280:. This is trivial, as
103:
1708:
1659:
1610:
1561:
1512:
1463:
1437:
1396:
1307:
1140:
1063:
970:
844:
770:
714:
631:
585:
447:
288:
178:indeterminates over
64:
24:refers usually to a
1779:Field (mathematics)
1763:Generic Polynomials
1384:for a finite group
567:Generic Polynomial
1742:
1693:
1644:
1595:
1546:
1497:
1443:
1415:
1357:
1280:
1113:
1036:
943:
817:
748:
687:
604:
467:
349:
224:indeterminates to
212:with Galois group
124:generic polynomial
120:generic polynomial
98:
36:. For example, if
22:generic polynomial
1382:generic dimension
1376:Generic dimension
1370:
1369:
492:alternating group
442:Heisenberg groups
1791:
1751:
1749:
1748:
1743:
1735:
1734:
1725:
1724:
1723:
1702:
1700:
1699:
1694:
1686:
1685:
1676:
1675:
1674:
1653:
1651:
1650:
1645:
1637:
1636:
1627:
1626:
1625:
1604:
1602:
1601:
1596:
1588:
1587:
1578:
1577:
1576:
1555:
1553:
1552:
1547:
1539:
1538:
1529:
1528:
1527:
1506:
1504:
1503:
1498:
1490:
1489:
1480:
1479:
1478:
1452:
1450:
1449:
1444:
1424:
1422:
1421:
1416:
1411:
1410:
1366:
1364:
1363:
1358:
1335:
1334:
1319:
1318:
1289:
1287:
1286:
1281:
1264:
1263:
1230:
1229:
1214:
1213:
1180:
1179:
1152:
1151:
1122:
1120:
1119:
1114:
1091:
1090:
1075:
1074:
1045:
1043:
1042:
1037:
1017:
1016:
1004:
1003:
982:
981:
952:
950:
949:
944:
933:
932:
920:
919:
910:
909:
897:
896:
878:
877:
856:
855:
826:
824:
823:
818:
807:
806:
785:
784:
757:
755:
754:
749:
726:
725:
696:
694:
693:
688:
659:
658:
643:
642:
613:
611:
610:
605:
597:
596:
561:
476:
474:
473:
468:
466:
465:
464:
463:
430:quaternion group
358:
356:
355:
350:
348:
347:
329:
328:
313:
312:
300:
299:
259:of order eight.
182:, such that the
142:monic polynomial
107:
105:
104:
99:
79:
78:
59:
53:
47:
41:
1799:
1798:
1794:
1793:
1792:
1790:
1789:
1788:
1769:
1768:
1758:
1726:
1714:
1706:
1705:
1677:
1665:
1657:
1656:
1628:
1616:
1608:
1607:
1579:
1567:
1559:
1558:
1530:
1518:
1510:
1509:
1481:
1469:
1461:
1460:
1435:
1434:
1402:
1394:
1393:
1378:
1326:
1310:
1305:
1304:
1301:
1255:
1221:
1205:
1171:
1143:
1138:
1137:
1134:
1082:
1066:
1061:
1060:
1057:
1008:
995:
973:
968:
967:
964:
924:
911:
901:
888:
869:
847:
842:
841:
838:
798:
776:
768:
767:
717:
712:
711:
708:
650:
634:
629:
628:
625:
588:
583:
582:
579:
559:
537:
530:
523:
509:
499:
455:
450:
445:
444:
437:
420:
385:
371:
339:
314:
304:
291:
286:
285:
279:
270:symmetric group
265:
184:splitting field
173:
164:
70:
62:
61:
55:
49:
43:
37:
12:
11:
5:
1797:
1795:
1787:
1786:
1781:
1771:
1770:
1767:
1766:
1757:
1754:
1753:
1752:
1741:
1738:
1733:
1729:
1722:
1717:
1713:
1703:
1692:
1689:
1684:
1680:
1673:
1668:
1664:
1654:
1643:
1640:
1635:
1631:
1624:
1619:
1615:
1605:
1594:
1591:
1586:
1582:
1575:
1570:
1566:
1556:
1545:
1542:
1537:
1533:
1526:
1521:
1517:
1507:
1496:
1493:
1488:
1484:
1477:
1472:
1468:
1442:
1414:
1409:
1405:
1401:
1377:
1374:
1368:
1367:
1356:
1353:
1350:
1347:
1344:
1341:
1338:
1333:
1329:
1325:
1322:
1317:
1313:
1302:
1297:
1291:
1290:
1279:
1276:
1273:
1270:
1267:
1262:
1258:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1228:
1224:
1220:
1217:
1212:
1208:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1178:
1174:
1170:
1167:
1164:
1161:
1158:
1155:
1150:
1146:
1135:
1130:
1124:
1123:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1089:
1085:
1081:
1078:
1073:
1069:
1058:
1053:
1047:
1046:
1035:
1032:
1029:
1026:
1023:
1020:
1015:
1011:
1007:
1002:
998:
994:
991:
988:
985:
980:
976:
965:
960:
954:
953:
942:
939:
936:
931:
927:
923:
918:
914:
908:
904:
900:
895:
891:
887:
884:
881:
876:
872:
868:
865:
862:
859:
854:
850:
839:
834:
828:
827:
816:
813:
810:
805:
801:
797:
794:
791:
788:
783:
779:
775:
765:
759:
758:
747:
744:
741:
738:
735:
732:
729:
724:
720:
709:
704:
698:
697:
686:
683:
680:
677:
674:
671:
668:
665:
662:
657:
653:
649:
646:
641:
637:
626:
621:
615:
614:
603:
600:
595:
591:
580:
575:
569:
568:
565:
558:
555:
554:
553:
550:wreath product
546:
543:direct product
539:
535:
528:
521:
511:
507:
501:
497:
488:
462:
458:
453:
439:
435:
426:
416:
411:dihedral group
407:
381:
377:Cyclic groups
374:
373:
367:
360:
359:
346:
342:
338:
335:
332:
327:
324:
321:
317:
311:
307:
303:
298:
294:
282:
281:
275:
264:
261:
169:
162:
114:, a branch of
97:
94:
91:
88:
85:
82:
77:
73:
69:
34:indeterminates
13:
10:
9:
6:
4:
3:
2:
1796:
1785:
1784:Galois theory
1782:
1780:
1777:
1776:
1774:
1764:
1760:
1759:
1755:
1739:
1736:
1731:
1727:
1715:
1711:
1704:
1690:
1687:
1682:
1678:
1666:
1662:
1655:
1641:
1638:
1633:
1629:
1617:
1613:
1606:
1592:
1589:
1584:
1580:
1568:
1564:
1557:
1543:
1540:
1535:
1531:
1519:
1515:
1508:
1494:
1491:
1486:
1482:
1470:
1466:
1459:
1458:
1457:
1454:
1432:
1428:
1412:
1407:
1403:
1399:
1391:
1388:over a field
1387:
1383:
1375:
1373:
1351:
1348:
1345:
1339:
1336:
1331:
1327:
1323:
1320:
1315:
1311:
1303:
1300:
1296:
1293:
1292:
1277:
1274:
1271:
1268:
1265:
1260:
1256:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1226:
1222:
1215:
1210:
1206:
1199:
1196:
1193:
1190:
1187:
1181:
1176:
1172:
1165:
1162:
1159:
1153:
1148:
1144:
1136:
1133:
1129:
1126:
1125:
1107:
1104:
1101:
1095:
1092:
1087:
1083:
1079:
1076:
1071:
1067:
1059:
1056:
1052:
1049:
1048:
1030:
1027:
1024:
1018:
1013:
1009:
1005:
1000:
996:
992:
989:
986:
983:
978:
974:
966:
963:
959:
956:
955:
937:
934:
929:
925:
916:
912:
906:
902:
898:
893:
889:
882:
879:
874:
870:
863:
860:
857:
852:
848:
840:
837:
833:
830:
829:
811:
808:
803:
799:
789:
786:
781:
777:
766:
764:
761:
760:
742:
739:
736:
730:
727:
722:
718:
710:
707:
703:
700:
699:
684:
681:
678:
672:
669:
666:
660:
655:
651:
647:
644:
639:
635:
627:
624:
620:
617:
616:
601:
598:
593:
589:
581:
578:
574:
571:
570:
566:
563:
562:
556:
551:
547:
544:
540:
534:
527:
520:
516:
512:
506:
502:
496:
493:
489:
486:
483:
480:
460:
456:
451:
443:
440:
434:
431:
427:
424:
419:
415:
412:
408:
405:
401:
397:
393:
389:
384:
380:
376:
375:
370:
366:
362:
361:
344:
340:
336:
333:
330:
325:
322:
319:
315:
309:
305:
301:
296:
292:
284:
283:
278:
274:
271:
267:
266:
262:
260:
258:
254:
249:
247:
243:
239:
235:
231:
227:
223:
219:
215:
211:
207:
203:
199:
196:
192:
188:
185:
181:
177:
172:
168:
161:
157:
153:
150:
146:
143:
139:
136:
132:
129:
125:
121:
117:
113:
112:Galois theory
108:
95:
92:
89:
86:
83:
80:
75:
71:
67:
58:
52:
46:
40:
35:
31:
27:
23:
19:
1762:
1756:Publications
1455:
1430:
1426:
1389:
1385:
1381:
1379:
1371:
1298:
1294:
1131:
1127:
1054:
1050:
961:
957:
835:
831:
762:
705:
701:
622:
618:
576:
572:
532:
525:
518:
514:
504:
494:
484:
432:
422:
417:
413:
403:
399:
387:
382:
378:
368:
364:
276:
272:
257:cyclic group
250:
245:
241:
237:
233:
229:
228:elements of
225:
221:
217:
213:
209:
205:
201:
197:
195:Galois group
190:
186:
179:
175:
170:
166:
159:
155:
151:
144:
137:
130:
128:finite group
123:
119:
109:
56:
50:
44:
38:
30:coefficients
21:
15:
110:However in
18:mathematics
1773:Categories
1456:Examples:
1392:, denoted
394:by eight.
26:polynomial
1441:∞
1337:−
1247:−
1238:−
1232:−
1191:−
1163:−
1093:−
1028:−
984:−
858:−
809:−
787:−
728:−
670:−
645:−
599:−
392:divisible
334:⋯
323:−
234:F-generic
477:for any
386:, where
396:Lenstra
390:is not
246:generic
165:, ...,
116:algebra
531:, and
133:and a
126:for a
48:, and
28:whose
1433:, or
1429:over
564:Group
482:prime
200:over
174:) in
140:is a
135:field
1380:The
490:The
428:The
268:The
240:; a
193:has
32:are
20:, a
479:odd
189:of
60:is
16:In
1775::
524:,
154:=
42:,
1740:2
1737:=
1732:5
1728:S
1721:Q
1716:d
1712:g
1691:2
1688:=
1683:5
1679:D
1672:Q
1667:d
1663:g
1642:2
1639:=
1634:4
1630:S
1623:Q
1618:d
1614:g
1593:2
1590:=
1585:4
1581:D
1574:Q
1569:d
1565:g
1544:1
1541:=
1536:3
1532:S
1525:Q
1520:d
1516:g
1495:1
1492:=
1487:3
1483:A
1476:Q
1471:d
1467:g
1431:F
1427:G
1413:G
1408:F
1404:d
1400:g
1390:F
1386:G
1355:)
1352:1
1349:+
1346:x
1343:(
1340:t
1332:3
1328:x
1324:s
1321:+
1316:5
1312:x
1299:5
1295:S
1278:t
1275:+
1272:x
1269:s
1266:+
1261:2
1257:x
1253:)
1250:1
1244:s
1241:2
1235:t
1227:2
1223:t
1219:(
1216:+
1211:3
1207:x
1203:)
1200:3
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