954:
1426:
1613:
872:
1653:
1354:
1053:, and that two such irreducible fractions are equal if and only if one may pass from one to the other by multiplying the numerator and the denominator by the same non zero element of
1481:
1301:
1193:
1545:
793:
684:
636:
867:
129:
1511:
449:
410:
241:
507:
1245:
1221:
1122:
1078:
1027:
980:
601:
280:
338:
734:
1741:
1093:
354:
1367:
828:
1573:
1729:
364:
248:
1618:
28:
1768:
748:
51:
40:
1318:
558:
368:
1447:
1254:
1146:
949:{\displaystyle {\begin{aligned}\varphi :K&\rightarrow L\\f(X)&\mapsto f(\theta )\,.\end{aligned}}}
701:
697:
341:
195:
1520:
753:
651:
606:
1046:
796:
1101:
96:
1492:
987:
203:
425:
386:
217:
457:
1737:
1230:
1206:
1107:
1063:
1012:
965:
859:
570:
291:
139:
1747:
1683:
1089:
309:
244:
55:
253:
1751:
1733:
1133:
852:
314:
36:
710:
17:
1762:
1141:
452:
1721:
1551:
824:
550:
380:
1248:
554:
1224:
207:
983:
47:. Simple extensions are well understood and can be completely classified.
1125:
1669:
a field of rational functions, is generated by the formal variable
815:
To distinguish these meanings, one uses the term "generator" or
1421:{\displaystyle 1,\theta ,\theta ^{2},\ldots ,\theta ^{n-1}}
819:
for the weaker meaning, reserving "primitive element" or
181:
There are two different kinds of simple extensions (see
1686:
for the multiplication map on a simple field extension
743:
is usually reserved for a stronger notion, an element
1621:
1608:{\displaystyle \mathbf {Q} ({\sqrt {3}},{\sqrt {7}})}
1576:
1523:
1495:
1450:
1370:
1321:
1257:
1233:
1209:
1149:
1110:
1066:
1015:
968:
870:
756:
713:
654:
609:
573:
460:
428:
389:
317:
256:
220:
99:
162:by the field operations +, −, •, / . Equivalently,
1647:
1607:
1539:
1505:
1475:
1420:
1348:
1295:
1239:
1215:
1187:
1116:
1072:
1021:
974:
948:
787:
728:
678:
630:
595:
501:
443:
404:
332:
274:
235:
123:
1140:is an irreducible polynomial, and thus that the
739:However, in the case of finite fields, the term
1648:{\displaystyle \theta ={\sqrt {3}}+{\sqrt {7}}}
412:can be uniquely expressed as a polynomial in
8:
1290:
1275:
1182:
1167:
825:Finite field § Multiplicative structure
782:
776:
858:, one of its main properties is the unique
1349:{\displaystyle n=\operatorname {deg} p(X)}
1638:
1628:
1620:
1595:
1585:
1577:
1575:
1530:
1522:
1496:
1494:
1463:
1449:
1406:
1387:
1369:
1320:
1270:
1256:
1232:
1208:
1162:
1148:
1109:
1065:
1014:
967:
938:
871:
869:
761:
755:
712:
667:
663:
662:
653:
644:elements is a simple extension of degree
622:
618:
617:
608:
584:
572:
473:
459:
427:
388:
316:
255:
219:
166:is the smallest field that contains both
98:
986:, it may be extended injectively to the
1696:
690:is generated as a field by any element
1476:{\displaystyle \theta =i={\sqrt {-1}}}
1296:{\displaystyle K/\langle p(X)\rangle }
1188:{\displaystyle K/\langle p(X)\rangle }
1704:
1307:. This implies that every element of
1041:. This implies that every element of
7:
1540:{\displaystyle \theta ={\sqrt {2}}}
1311:is equal to a unique polynomial in
799:, so that every nonzero element of
788:{\displaystyle L^{\times }=L-\{0\}}
679:{\displaystyle K=\mathbb {F} _{p}.}
54:provides a characterization of the
631:{\displaystyle L=\mathbb {F} _{q}}
25:
811:using only the group operation •
525:for the extension; one says also
383:. In this case, every element of
134:This means that every element of
1578:
1315:of degree lower than the degree
829:Primitive element (finite field)
823:for the stronger meaning. (See
304:is a root of a polynomial over
202:, which means that it is not a
152:; that is, it is produced from
1602:
1582:
1343:
1337:
1287:
1281:
1267:
1261:
1179:
1173:
1159:
1153:
935:
929:
923:
916:
910:
897:
890:
884:
835:Structure of simple extensions
723:
717:
493:
490:
484:
478:
470:
464:
438:
432:
399:
393:
327:
321:
266:
260:
230:
224:
183:Structure of simple extensions
115:
109:
43:of a single element, called a
1:
1730:Graduate Texts in Mathematics
1554:(i.e., a finite extension of
553:is a simple extension of the
365:degree of the field extension
124:{\displaystyle L=K(\theta ).}
1732:. Vol. 158. New York:
1506:{\displaystyle {\sqrt {2}}}
511:In both cases, the element
249:field of rational functions
80:if there exists an element
1785:
444:{\displaystyle K(\theta )}
405:{\displaystyle K(\theta )}
236:{\displaystyle K(\theta )}
843:be a simple extension of
502:{\displaystyle K/(p(X)).}
353:as a root, is called the
52:primitive element theorem
39:that is generated by the
1558:) is a simple extension
1240:{\displaystyle \varphi }
1216:{\displaystyle \varphi }
1117:{\displaystyle \varphi }
1088:) be a generator of its
1073:{\displaystyle \varphi }
1022:{\displaystyle \varphi }
975:{\displaystyle \varphi }
807:, i.e. is produced from
596:{\displaystyle q=p^{n},}
363:. Its degree equals the
1029:is an isomorphism from
821:group primitive element
817:field primitive element
148:, with coefficients in
1649:
1609:
1541:
1507:
1477:
1422:
1350:
1297:
1241:
1217:
1189:
1118:
1080:is not injective, let
1074:
1023:
976:
950:
789:
730:
698:irreducible polynomial
680:
632:
597:
567:is a prime number and
503:
445:
406:
334:
276:
237:
138:can be expressed as a
125:
18:Simple field extension
1650:
1610:
1542:
1508:
1478:
1423:
1356:. That is, we have a
1351:
1298:
1242:
1218:
1190:
1119:
1075:
1024:
977:
959:Two cases may occur.
951:
790:
731:
696:that is a root of an
681:
633:
598:
561:. More precisely, if
504:
451:is isomorphic to the
446:
407:
335:
277:
275:{\displaystyle K(X).}
238:
210:with coefficients in
126:
1619:
1574:
1521:
1493:
1448:
1368:
1319:
1255:
1231:
1207:
1147:
1136:. This implies that
1108:
1092:, which is thus the
1064:
1047:irreducible fraction
1013:
1009:, this implies that
966:
868:
797:multiplicative group
754:
711:
652:
607:
571:
458:
426:
418:of degree less than
387:
333:{\displaystyle p(X)}
315:
254:
218:
97:
549:For example, every
58:simple extensions.
1645:
1605:
1537:
1503:
1473:
1418:
1346:
1293:
1237:
1213:
1185:
1114:
1094:minimal polynomial
1070:
1049:of polynomials in
1019:
988:field of fractions
972:
946:
944:
785:
726:
676:
628:
593:
519:generating element
499:
441:
402:
355:minimal polynomial
330:
272:
233:
121:
66:A field extension
1643:
1633:
1600:
1590:
1535:
1501:
1471:
860:ring homomorphism
741:primitive element
729:{\displaystyle K}
523:primitive element
140:rational fraction
45:primitive element
16:(Redirected from
1776:
1769:Field extensions
1755:
1708:
1701:
1684:Companion matrix
1654:
1652:
1651:
1646:
1644:
1639:
1634:
1629:
1615:is generated by
1614:
1612:
1611:
1606:
1601:
1596:
1591:
1586:
1581:
1546:
1544:
1543:
1538:
1536:
1531:
1512:
1510:
1509:
1504:
1502:
1497:
1482:
1480:
1479:
1474:
1472:
1464:
1427:
1425:
1424:
1419:
1417:
1416:
1392:
1391:
1355:
1353:
1352:
1347:
1302:
1300:
1299:
1294:
1274:
1246:
1244:
1243:
1238:
1222:
1220:
1219:
1214:
1199:is generated by
1194:
1192:
1191:
1186:
1166:
1123:
1121:
1120:
1115:
1079:
1077:
1076:
1071:
1028:
1026:
1025:
1020:
1005:is generated by
981:
979:
978:
973:
955:
953:
952:
947:
945:
794:
792:
791:
786:
766:
765:
735:
733:
732:
727:
695:
685:
683:
682:
677:
672:
671:
666:
643:
637:
635:
634:
629:
627:
626:
621:
602:
600:
599:
594:
589:
588:
566:
545:
539:
530:
516:
508:
506:
505:
500:
477:
450:
448:
447:
442:
421:
417:
411:
409:
408:
403:
378:
374:
362:
352:
346:
339:
337:
336:
331:
310:monic polynomial
307:
303:
297:
289:
281:
279:
278:
273:
242:
240:
239:
234:
213:
201:
193:
177:
170:
165:
161:
158:and elements of
157:
151:
147:
137:
130:
128:
127:
122:
85:
78:simple extension
75:
33:simple extension
21:
1784:
1783:
1779:
1778:
1777:
1775:
1774:
1773:
1759:
1758:
1744:
1734:Springer-Verlag
1720:
1717:
1712:
1711:
1702:
1698:
1693:
1680:
1617:
1616:
1572:
1571:
1570:. For example,
1519:
1518:
1491:
1490:
1446:
1445:
1434:
1402:
1383:
1366:
1365:
1317:
1316:
1253:
1252:
1229:
1228:
1205:
1204:
1195:is a field. As
1145:
1144:
1134:integral domain
1106:
1105:
1062:
1061:
1045:is equal to an
1011:
1010:
964:
963:
943:
942:
919:
904:
903:
893:
866:
865:
853:polynomial ring
837:
757:
752:
751:
709:
708:
691:
661:
650:
649:
639:
616:
605:
604:
580:
569:
568:
562:
541:
535:
526:
512:
456:
455:
424:
423:
419:
413:
385:
384:
376:
372:
367:, that is, the
358:
348:
344:
313:
312:
305:
299:
295:
285:
252:
251:
216:
215:
214:. In this case
211:
199:
189:
173:
168:
163:
159:
153:
149:
143:
135:
95:
94:
81:
67:
64:
37:field extension
23:
22:
15:
12:
11:
5:
1782:
1780:
1772:
1771:
1761:
1760:
1757:
1756:
1742:
1716:
1713:
1710:
1709:
1695:
1694:
1692:
1689:
1688:
1687:
1679:
1676:
1675:
1674:
1656:
1642:
1637:
1632:
1627:
1624:
1604:
1599:
1594:
1589:
1584:
1580:
1548:
1534:
1529:
1526:
1500:
1484:
1470:
1467:
1462:
1459:
1456:
1453:
1433:
1430:
1415:
1412:
1409:
1405:
1401:
1398:
1395:
1390:
1386:
1382:
1379:
1376:
1373:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1292:
1289:
1286:
1283:
1280:
1277:
1273:
1269:
1266:
1263:
1260:
1236:
1212:
1184:
1181:
1178:
1175:
1172:
1169:
1165:
1161:
1158:
1155:
1152:
1132:, and thus an
1113:
1069:
1018:
971:
957:
956:
941:
937:
934:
931:
928:
925:
922:
920:
918:
915:
912:
909:
906:
905:
902:
899:
896:
894:
892:
889:
886:
883:
880:
877:
874:
873:
836:
833:
803:is a power of
784:
781:
778:
775:
772:
769:
764:
760:
725:
722:
719:
716:
675:
670:
665:
660:
657:
625:
620:
615:
612:
592:
587:
583:
579:
576:
559:characteristic
533:generated over
498:
495:
492:
489:
486:
483:
480:
476:
472:
469:
466:
463:
440:
437:
434:
431:
401:
398:
395:
392:
329:
326:
323:
320:
271:
268:
265:
262:
259:
232:
229:
226:
223:
196:transcendental
132:
131:
120:
117:
114:
111:
108:
105:
102:
63:
60:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1781:
1770:
1767:
1766:
1764:
1753:
1749:
1745:
1743:0-387-94408-7
1739:
1735:
1731:
1727:
1723:
1722:Roman, Steven
1719:
1718:
1714:
1706:
1700:
1697:
1690:
1685:
1682:
1681:
1677:
1672:
1668:
1664:
1660:
1657:
1640:
1635:
1630:
1625:
1622:
1597:
1592:
1587:
1569:
1565:
1561:
1557:
1553:
1549:
1532:
1527:
1524:
1517:generated by
1516:
1498:
1488:
1485:
1468:
1465:
1460:
1457:
1454:
1451:
1444:generated by
1443:
1439:
1436:
1435:
1431:
1429:
1413:
1410:
1407:
1403:
1399:
1396:
1393:
1388:
1384:
1380:
1377:
1374:
1371:
1363:
1359:
1340:
1334:
1331:
1328:
1325:
1322:
1314:
1310:
1306:
1284:
1278:
1271:
1264:
1258:
1250:
1234:
1226:
1210:
1202:
1198:
1176:
1170:
1163:
1156:
1150:
1143:
1142:quotient ring
1139:
1135:
1131:
1127:
1111:
1103:
1099:
1095:
1091:
1087:
1083:
1067:
1058:
1056:
1052:
1048:
1044:
1040:
1036:
1032:
1016:
1008:
1004:
1000:
996:
992:
989:
985:
969:
960:
939:
932:
926:
921:
913:
907:
900:
895:
887:
881:
878:
875:
864:
863:
862:
861:
857:
854:
850:
847:generated by
846:
842:
834:
832:
830:
826:
822:
818:
814:
810:
806:
802:
798:
779:
773:
770:
767:
762:
758:
750:
746:
742:
737:
720:
714:
706:
703:
699:
694:
689:
673:
668:
658:
655:
647:
642:
623:
613:
610:
590:
585:
581:
577:
574:
565:
560:
556:
552:
547:
544:
538:
534:
529:
524:
520:
515:
509:
496:
487:
481:
474:
467:
461:
454:
453:quotient ring
435:
429:
416:
396:
390:
382:
370:
366:
361:
356:
351:
343:
324:
318:
311:
302:
293:
288:
282:
269:
263:
257:
250:
246:
227:
221:
209:
205:
197:
192:
186:
184:
179:
176:
171:
156:
146:
141:
118:
112:
106:
103:
100:
93:
92:
91:
89:
84:
79:
74:
70:
61:
59:
57:
53:
48:
46:
42:
38:
34:
30:
19:
1726:Field Theory
1725:
1699:
1670:
1666:
1662:
1658:
1567:
1563:
1559:
1555:
1552:number field
1514:
1486:
1441:
1437:
1361:
1357:
1312:
1308:
1304:
1200:
1196:
1137:
1129:
1097:
1085:
1081:
1059:
1054:
1050:
1042:
1038:
1034:
1030:
1006:
1002:
998:
994:
990:
961:
958:
855:
848:
844:
840:
838:
820:
816:
812:
808:
804:
800:
744:
740:
738:
704:
692:
687:
645:
640:
563:
557:of the same
551:finite field
548:
542:
536:
532:
527:
522:
518:
517:is called a
513:
510:
414:
381:vector space
375:viewed as a
359:
349:
300:
286:
283:
190:
188:The element
187:
182:
180:
174:
167:
154:
144:
133:
87:
82:
77:
76:is called a
72:
68:
65:
49:
44:
32:
29:field theory
26:
1566:) for some
1249:isomorphism
1247:induces an
555:prime field
340:of minimal
298:; that is,
284:Otherwise,
1752:0816.12001
1715:Literature
1705:Roman 1995
1691:References
1225:surjective
851:. For the
603:the field
245:isomorphic
208:polynomial
62:Definition
41:adjunction
1623:θ
1525:θ
1466:−
1452:θ
1411:−
1404:θ
1397:…
1385:θ
1378:θ
1364:given by
1360:basis of
1332:
1291:⟩
1276:⟨
1235:φ
1211:φ
1183:⟩
1168:⟨
1112:φ
1068:φ
1017:φ
984:injective
970:φ
933:θ
924:↦
898:→
876:φ
774:−
763:×
749:generates
686:In fact,
436:θ
397:θ
369:dimension
292:algebraic
228:θ
113:θ
1763:Category
1724:(1995).
1678:See also
1432:Examples
1001:. Since
185:below).
1126:subring
1037:) onto
347:, with
247:to the
206:of any
194:may be
1750:
1740:
1227:, and
1100:. The
1090:kernel
702:degree
422:, and
342:degree
308:. The
56:finite
1303:onto
1251:from
1124:is a
1102:image
997:) of
831:).
795:as a
747:that
294:over
198:over
90:with
35:is a
1738:ISBN
1665:) /
1550:Any
1513:) /
839:Let
827:and
736:.
204:root
172:and
50:The
31:, a
1748:Zbl
1428:.
1329:deg
1223:is
1128:of
1104:of
1096:of
1060:If
982:is
962:If
707:in
700:of
648:of
638:of
540:by
531:is
521:or
371:of
357:of
290:is
243:is
142:in
86:in
27:In
1765::
1746:.
1736:.
1728:.
1667:F,
1440:/
1358:K-
1203:,
1057:.
546:.
178:.
1754:.
1707:)
1703:(
1673:.
1671:X
1663:X
1661:(
1659:F
1655:.
1641:7
1636:+
1631:3
1626:=
1603:)
1598:7
1593:,
1588:3
1583:(
1579:Q
1568:θ
1564:θ
1562:(
1560:Q
1556:Q
1547:.
1533:2
1528:=
1515:Q
1499:2
1489:(
1487:Q
1483:.
1469:1
1461:=
1458:i
1455:=
1442:R
1438:C
1414:1
1408:n
1400:,
1394:,
1389:2
1381:,
1375:,
1372:1
1362:L
1344:)
1341:X
1338:(
1335:p
1326:=
1323:n
1313:θ
1309:L
1305:L
1288:)
1285:X
1282:(
1279:p
1272:/
1268:]
1265:X
1262:[
1259:K
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1177:X
1174:(
1171:p
1164:/
1160:]
1157:X
1154:[
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1082:p
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993:(
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940:.
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888:X
885:[
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809:γ
805:γ
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783:}
780:0
777:{
771:L
768:=
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745:γ
724:]
721:X
718:[
715:K
705:n
693:θ
688:L
674:.
669:p
664:F
659:=
656:K
646:n
641:q
624:q
619:F
614:=
611:L
591:,
586:n
582:p
578:=
575:q
564:p
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497:.
494:)
491:)
488:X
485:(
482:p
479:(
475:/
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225:(
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119:.
116:)
110:(
107:K
104:=
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73:K
71:/
69:L
20:)
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