943:
1415:
1602:
861:
1642:
1343:
1042:, 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
1470:
1290:
1182:
1534:
782:
673:
625:
856:
118:
1500:
438:
399:
230:
496:
1234:
1210:
1111:
1067:
1016:
969:
590:
269:
327:
723:
1730:
1082:
343:
1356:
817:
1562:
1718:
353:
237:
1607:
17:
1757:
737:
40:
29:
1307:
547:
357:
1436:
1243:
1135:
938:{\displaystyle {\begin{aligned}\varphi :K&\rightarrow L\\f(X)&\mapsto f(\theta )\,.\end{aligned}}}
690:
686:
330:
184:
1509:
742:
640:
595:
1035:
785:
1090:
85:
1481:
976:
192:
414:
375:
206:
446:
1726:
1219:
1195:
1096:
1052:
1001:
954:
848:
559:
280:
128:
1736:
1672:
1078:
298:
233:
44:
242:
1740:
1722:
1122:
841:
303:
25:
699:
1751:
1130:
441:
1710:
1540:
813:
539:
369:
1237:
543:
1213:
196:
972:
36:. Simple extensions are well understood and can be completely classified.
1114:
1658:
a field of rational functions, is generated by the formal variable
804:
To distinguish these meanings, one uses the term "generator" or
1410:{\displaystyle 1,\theta ,\theta ^{2},\ldots ,\theta ^{n-1}}
808:
for the weaker meaning, reserving "primitive element" or
170:
There are two different kinds of simple extensions (see
1675:
for the multiplication map on a simple field extension
732:
is usually reserved for a stronger notion, an element
1610:
1597:{\displaystyle \mathbf {Q} ({\sqrt {3}},{\sqrt {7}})}
1565:
1512:
1484:
1439:
1359:
1310:
1246:
1222:
1198:
1138:
1099:
1055:
1004:
957:
859:
745:
702:
643:
598:
562:
449:
417:
378:
306:
245:
209:
88:
151:by the field operations +, −, •, / . Equivalently,
1636:
1596:
1528:
1494:
1464:
1409:
1337:
1284:
1228:
1204:
1176:
1105:
1061:
1010:
963:
937:
776:
717:
667:
619:
584:
490:
432:
393:
321:
263:
224:
112:
1129:is an irreducible polynomial, and thus that the
728:However, in the case of finite fields, the term
1637:{\displaystyle \theta ={\sqrt {3}}+{\sqrt {7}}}
401:can be uniquely expressed as a polynomial in
8:
1279:
1264:
1171:
1156:
814:Finite field § Multiplicative structure
771:
765:
847:, one of its main properties is the unique
1338:{\displaystyle n=\operatorname {deg} p(X)}
1627:
1617:
1609:
1584:
1574:
1566:
1564:
1519:
1511:
1485:
1483:
1452:
1438:
1395:
1376:
1358:
1309:
1259:
1245:
1221:
1197:
1151:
1137:
1098:
1054:
1003:
956:
927:
860:
858:
750:
744:
701:
656:
652:
651:
642:
633:elements is a simple extension of degree
611:
607:
606:
597:
573:
561:
462:
448:
416:
377:
305:
244:
208:
155:is the smallest field that contains both
87:
975:, it may be extended injectively to the
1685:
679:is generated as a field by any element
1465:{\displaystyle \theta =i={\sqrt {-1}}}
1285:{\displaystyle K/\langle p(X)\rangle }
1177:{\displaystyle K/\langle p(X)\rangle }
1693:
1296:. This implies that every element of
1030:. This implies that every element of
7:
1529:{\displaystyle \theta ={\sqrt {2}}}
1300:is equal to a unique polynomial in
788:, so that every nonzero element of
777:{\displaystyle L^{\times }=L-\{0\}}
668:{\displaystyle K=\mathbb {F} _{p}.}
43:provides a characterization of the
620:{\displaystyle L=\mathbb {F} _{q}}
14:
800:using only the group operation •
514:for the extension; one says also
372:. In this case, every element of
123:This means that every element of
1567:
1304:of degree lower than the degree
818:Primitive element (finite field)
812:for the stronger meaning. (See
293:is a root of a polynomial over
191:, which means that it is not a
141:; that is, it is produced from
1591:
1571:
1332:
1326:
1276:
1270:
1256:
1250:
1168:
1162:
1148:
1142:
924:
918:
912:
905:
899:
886:
879:
873:
824:Structure of simple extensions
712:
706:
482:
479:
473:
467:
459:
453:
427:
421:
388:
382:
316:
310:
255:
249:
219:
213:
172:Structure of simple extensions
104:
98:
32:of a single element, called a
1:
1719:Graduate Texts in Mathematics
1543:(i.e., a finite extension of
542:is a simple extension of the
354:degree of the field extension
113:{\displaystyle L=K(\theta ).}
1721:. Vol. 158. New York:
1495:{\displaystyle {\sqrt {2}}}
500:In both cases, the element
238:field of rational functions
69:if there exists an element
1774:
433:{\displaystyle K(\theta )}
394:{\displaystyle K(\theta )}
225:{\displaystyle K(\theta )}
832:be a simple extension of
491:{\displaystyle K/(p(X)).}
342:as a root, is called the
41:primitive element theorem
28:that is generated by the
1547:) is a simple extension
1229:{\displaystyle \varphi }
1205:{\displaystyle \varphi }
1106:{\displaystyle \varphi }
1077:) be a generator of its
1062:{\displaystyle \varphi }
1011:{\displaystyle \varphi }
964:{\displaystyle \varphi }
796:, i.e. is produced from
585:{\displaystyle q=p^{n},}
352:. Its degree equals the
1018:is an isomorphism from
810:group primitive element
806:field primitive element
137:, with coefficients in
1638:
1598:
1530:
1496:
1466:
1411:
1339:
1286:
1230:
1206:
1178:
1107:
1069:is not injective, let
1063:
1012:
965:
939:
778:
719:
687:irreducible polynomial
669:
621:
586:
556:is a prime number and
492:
434:
395:
323:
265:
226:
127:can be expressed as a
114:
1639:
1599:
1531:
1497:
1467:
1412:
1345:. That is, we have a
1340:
1287:
1231:
1207:
1179:
1108:
1064:
1013:
966:
948:Two cases may occur.
940:
779:
720:
685:that is a root of an
670:
622:
587:
550:. More precisely, if
493:
440:is isomorphic to the
435:
396:
324:
266:
264:{\displaystyle K(X).}
227:
199:with coefficients in
115:
1608:
1563:
1510:
1482:
1437:
1357:
1308:
1244:
1220:
1196:
1136:
1125:. This implies that
1097:
1081:, which is thus the
1053:
1036:irreducible fraction
1002:
998:, this implies that
955:
857:
786:multiplicative group
743:
700:
641:
596:
560:
447:
415:
407:of degree less than
376:
322:{\displaystyle p(X)}
304:
243:
207:
86:
538:For example, every
47:simple extensions.
1634:
1594:
1526:
1492:
1462:
1407:
1335:
1282:
1226:
1202:
1174:
1103:
1083:minimal polynomial
1059:
1038:of polynomials in
1008:
977:field of fractions
961:
935:
933:
774:
715:
665:
617:
582:
508:generating element
488:
430:
391:
344:minimal polynomial
319:
261:
222:
110:
55:A field extension
1632:
1622:
1589:
1579:
1524:
1490:
1460:
849:ring homomorphism
730:primitive element
718:{\displaystyle K}
512:primitive element
129:rational fraction
34:primitive element
1765:
1758:Field extensions
1744:
1697:
1690:
1673:Companion matrix
1643:
1641:
1640:
1635:
1633:
1628:
1623:
1618:
1604:is generated by
1603:
1601:
1600:
1595:
1590:
1585:
1580:
1575:
1570:
1535:
1533:
1532:
1527:
1525:
1520:
1501:
1499:
1498:
1493:
1491:
1486:
1471:
1469:
1468:
1463:
1461:
1453:
1416:
1414:
1413:
1408:
1406:
1405:
1381:
1380:
1344:
1342:
1341:
1336:
1291:
1289:
1288:
1283:
1263:
1235:
1233:
1232:
1227:
1211:
1209:
1208:
1203:
1188:is generated by
1183:
1181:
1180:
1175:
1155:
1112:
1110:
1109:
1104:
1068:
1066:
1065:
1060:
1017:
1015:
1014:
1009:
994:is generated by
970:
968:
967:
962:
944:
942:
941:
936:
934:
783:
781:
780:
775:
755:
754:
724:
722:
721:
716:
684:
674:
672:
671:
666:
661:
660:
655:
632:
626:
624:
623:
618:
616:
615:
610:
591:
589:
588:
583:
578:
577:
555:
534:
528:
519:
505:
497:
495:
494:
489:
466:
439:
437:
436:
431:
410:
406:
400:
398:
397:
392:
367:
363:
351:
341:
335:
328:
326:
325:
320:
299:monic polynomial
296:
292:
286:
278:
270:
268:
267:
262:
231:
229:
228:
223:
202:
190:
182:
166:
159:
154:
150:
147:and elements of
146:
140:
136:
126:
119:
117:
116:
111:
74:
67:simple extension
64:
22:simple extension
1773:
1772:
1768:
1767:
1766:
1764:
1763:
1762:
1748:
1747:
1733:
1723:Springer-Verlag
1709:
1706:
1701:
1700:
1691:
1687:
1682:
1669:
1606:
1605:
1561:
1560:
1559:. For example,
1508:
1507:
1480:
1479:
1435:
1434:
1423:
1391:
1372:
1355:
1354:
1306:
1305:
1242:
1241:
1218:
1217:
1194:
1193:
1184:is a field. As
1134:
1133:
1123:integral domain
1095:
1094:
1051:
1050:
1034:is equal to an
1000:
999:
953:
952:
932:
931:
908:
893:
892:
882:
855:
854:
842:polynomial ring
826:
746:
741:
740:
698:
697:
680:
650:
639:
638:
628:
605:
594:
593:
569:
558:
557:
551:
530:
524:
515:
501:
445:
444:
413:
412:
408:
402:
374:
373:
365:
361:
356:, that is, the
347:
337:
333:
302:
301:
294:
288:
284:
274:
241:
240:
205:
204:
203:. In this case
200:
188:
178:
162:
157:
152:
148:
142:
138:
132:
124:
84:
83:
70:
56:
53:
26:field extension
12:
11:
5:
1771:
1769:
1761:
1760:
1750:
1749:
1746:
1745:
1731:
1705:
1702:
1699:
1698:
1684:
1683:
1681:
1678:
1677:
1676:
1668:
1665:
1664:
1663:
1645:
1631:
1626:
1621:
1616:
1613:
1593:
1588:
1583:
1578:
1573:
1569:
1537:
1523:
1518:
1515:
1489:
1473:
1459:
1456:
1451:
1448:
1445:
1442:
1422:
1419:
1404:
1401:
1398:
1394:
1390:
1387:
1384:
1379:
1375:
1371:
1368:
1365:
1362:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1281:
1278:
1275:
1272:
1269:
1266:
1262:
1258:
1255:
1252:
1249:
1225:
1201:
1173:
1170:
1167:
1164:
1161:
1158:
1154:
1150:
1147:
1144:
1141:
1121:, and thus an
1102:
1058:
1007:
960:
946:
945:
930:
926:
923:
920:
917:
914:
911:
909:
907:
904:
901:
898:
895:
894:
891:
888:
885:
883:
881:
878:
875:
872:
869:
866:
863:
862:
825:
822:
792:is a power of
773:
770:
767:
764:
761:
758:
753:
749:
714:
711:
708:
705:
664:
659:
654:
649:
646:
614:
609:
604:
601:
581:
576:
572:
568:
565:
548:characteristic
522:generated over
487:
484:
481:
478:
475:
472:
469:
465:
461:
458:
455:
452:
429:
426:
423:
420:
390:
387:
384:
381:
318:
315:
312:
309:
260:
257:
254:
251:
248:
221:
218:
215:
212:
185:transcendental
121:
120:
109:
106:
103:
100:
97:
94:
91:
52:
49:
13:
10:
9:
6:
4:
3:
2:
1770:
1759:
1756:
1755:
1753:
1742:
1738:
1734:
1732:0-387-94408-7
1728:
1724:
1720:
1716:
1712:
1711:Roman, Steven
1708:
1707:
1703:
1695:
1689:
1686:
1679:
1674:
1671:
1670:
1666:
1661:
1657:
1653:
1649:
1646:
1629:
1624:
1619:
1614:
1611:
1586:
1581:
1576:
1558:
1554:
1550:
1546:
1542:
1538:
1521:
1516:
1513:
1506:generated by
1505:
1487:
1477:
1474:
1457:
1454:
1449:
1446:
1443:
1440:
1433:generated by
1432:
1428:
1425:
1424:
1420:
1418:
1402:
1399:
1396:
1392:
1388:
1385:
1382:
1377:
1373:
1369:
1366:
1363:
1360:
1352:
1348:
1329:
1323:
1320:
1317:
1314:
1311:
1303:
1299:
1295:
1273:
1267:
1260:
1253:
1247:
1239:
1223:
1215:
1199:
1191:
1187:
1165:
1159:
1152:
1145:
1139:
1132:
1131:quotient ring
1128:
1124:
1120:
1116:
1100:
1092:
1088:
1084:
1080:
1076:
1072:
1056:
1047:
1045:
1041:
1037:
1033:
1029:
1025:
1021:
1005:
997:
993:
989:
985:
981:
978:
974:
958:
949:
928:
921:
915:
910:
902:
896:
889:
884:
876:
870:
867:
864:
853:
852:
851:
850:
846:
843:
839:
836:generated by
835:
831:
823:
821:
819:
815:
811:
807:
803:
799:
795:
791:
787:
768:
762:
759:
756:
751:
747:
739:
735:
731:
726:
709:
703:
695:
692:
688:
683:
678:
662:
657:
647:
644:
636:
631:
612:
602:
599:
579:
574:
570:
566:
563:
554:
549:
545:
541:
536:
533:
527:
523:
518:
513:
509:
504:
498:
485:
476:
470:
463:
456:
450:
443:
442:quotient ring
424:
418:
405:
385:
379:
371:
359:
355:
350:
345:
340:
332:
313:
307:
300:
291:
282:
277:
271:
258:
252:
246:
239:
235:
216:
210:
198:
194:
186:
181:
175:
173:
168:
165:
160:
145:
135:
130:
107:
101:
95:
92:
89:
82:
81:
80:
78:
73:
68:
63:
59:
50:
48:
46:
42:
37:
35:
31:
27:
23:
19:
1715:Field Theory
1714:
1688:
1659:
1655:
1651:
1647:
1556:
1552:
1548:
1544:
1541:number field
1503:
1475:
1430:
1426:
1350:
1346:
1301:
1297:
1293:
1189:
1185:
1126:
1118:
1086:
1074:
1070:
1048:
1043:
1039:
1031:
1027:
1023:
1019:
995:
991:
987:
983:
979:
950:
947:
844:
837:
833:
829:
827:
809:
805:
801:
797:
793:
789:
733:
729:
727:
693:
681:
676:
634:
629:
552:
546:of the same
540:finite field
537:
531:
525:
521:
516:
511:
507:
506:is called a
502:
499:
403:
370:vector space
364:viewed as a
348:
338:
289:
275:
272:
179:
177:The element
176:
171:
169:
163:
156:
143:
133:
122:
76:
71:
66:
65:is called a
61:
57:
54:
38:
33:
21:
18:field theory
15:
1555:) for some
1238:isomorphism
1236:induces an
544:prime field
329:of minimal
287:; that is,
273:Otherwise,
1741:0816.12001
1704:Literature
1694:Roman 1995
1680:References
1214:surjective
840:. For the
592:the field
234:isomorphic
197:polynomial
51:Definition
30:adjunction
1612:θ
1514:θ
1455:−
1441:θ
1400:−
1393:θ
1386:…
1374:θ
1367:θ
1353:given by
1349:basis of
1321:
1280:⟩
1265:⟨
1224:φ
1200:φ
1172:⟩
1157:⟨
1101:φ
1057:φ
1006:φ
973:injective
959:φ
922:θ
913:↦
887:→
865:φ
763:−
752:×
738:generates
675:In fact,
425:θ
386:θ
358:dimension
281:algebraic
217:θ
102:θ
1752:Category
1713:(1995).
1667:See also
1421:Examples
990:. Since
174:below).
1115:subring
1026:) onto
336:, with
236:to the
195:of any
183:may be
1739:
1729:
1216:, and
1089:. The
1079:kernel
691:degree
411:, and
331:degree
297:. The
45:finite
1292:onto
1240:from
1113:is a
1091:image
986:) of
820:).
784:as a
736:that
283:over
187:over
79:with
24:is a
1727:ISBN
1654:) /
1539:Any
1502:) /
828:Let
816:and
725:.
193:root
161:and
39:The
20:, a
1737:Zbl
1417:.
1318:deg
1212:is
1117:of
1093:of
1085:of
1049:If
971:is
951:If
696:in
689:of
637:of
627:of
529:by
520:is
510:or
360:of
346:of
279:is
232:is
131:in
75:in
16:In
1754::
1735:.
1725:.
1717:.
1656:F,
1429:/
1347:K-
1192:,
1046:.
535:.
167:.
1743:.
1696:)
1692:(
1662:.
1660:X
1652:X
1650:(
1648:F
1644:.
1630:7
1625:+
1620:3
1615:=
1592:)
1587:7
1582:,
1577:3
1572:(
1568:Q
1557:θ
1553:θ
1551:(
1549:Q
1545:Q
1536:.
1522:2
1517:=
1504:Q
1488:2
1478:(
1476:Q
1472:.
1458:1
1450:=
1447:i
1444:=
1431:R
1427:C
1403:1
1397:n
1389:,
1383:,
1378:2
1370:,
1364:,
1361:1
1351:L
1333:)
1330:X
1327:(
1324:p
1315:=
1312:n
1302:θ
1298:L
1294:L
1277:)
1274:X
1271:(
1268:p
1261:/
1257:]
1254:X
1251:[
1248:K
1190:θ
1186:L
1169:)
1166:X
1163:(
1160:p
1153:/
1149:]
1146:X
1143:[
1140:K
1127:p
1119:L
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1073:(
1071:p
1044:K
1040:θ
1032:L
1028:L
1024:X
1022:(
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996:θ
992:L
988:K
984:X
982:(
980:K
929:.
925:)
919:(
916:f
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897:f
890:L
880:]
877:X
874:[
871:K
868::
845:K
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830:L
802:.
798:γ
794:γ
790:L
772:}
769:0
766:{
760:L
757:=
748:L
734:γ
713:]
710:X
707:[
704:K
694:n
682:θ
677:L
663:.
658:p
653:F
648:=
645:K
635:n
630:q
613:q
608:F
603:=
600:L
580:,
575:n
571:p
567:=
564:q
553:p
532:θ
526:K
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503:θ
486:.
483:)
480:)
477:X
474:(
471:p
468:(
464:/
460:]
457:X
454:[
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428:)
422:(
419:K
409:n
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220:)
214:(
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189:K
180:θ
164:θ
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153:L
149:K
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134:θ
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108:.
105:)
99:(
96:K
93:=
90:L
77:L
72:θ
62:K
60:/
58:L
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