789:
342:
444:
633:
505:
1218:
641:
996:
572:
831:
386:
876:
546:
244:
There are many countable algebraically closed fields within the complex numbers, and strictly containing the field of algebraic numbers; these are the algebraic closures of
938:
1357:
1430:
1384:
294:
391:
577:
1112:. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of
1465:
449:
17:
1161:
784:{\displaystyle f_{\lambda }-\prod _{i=1}^{d}(x-u_{\lambda ,i})=\sum _{j=0}^{d-1}r_{\lambda ,j}\cdot x^{j}\in R}
219:
1376:
947:
551:
797:
1248:
358:
58:
1509:
1144:
93:
848:
518:
245:
916:
348:
1369:
Brawley, Joel V.; Schnibben, George E. (1989), "2.2 The
Algebraic Closure of a Finite Field",
1370:
1224:
62:
1421:. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd ed.).
1492:
1475:
1440:
1394:
1345:
1023:, because any polynomial with coefficients in this new field has its coefficients in some
8:
1253:
1086:
172:
50:
43:
1019:
can be constructed, etc. The union of all these extensions is the algebraic closure of
268:
202:
1461:
1426:
1380:
1258:
73:
1488:
1471:
1453:
1436:
1390:
1341:
1333:
345:
238:
35:
1460:. Chicago lectures in mathematics (Second ed.). University of Chicago Press.
1422:
1295:
1058:
838:
234:
190:
69:
1291:
1263:
227:
1503:
1337:
1136:
260:
256:
77:
286:
223:
89:
31:
1487:(Corrected reprint of the 2nd ed.). New York: Dover Publications.
1324:
Banaschewski, Bernhard (1992), "Algebraic closure without choice.",
1131:
The separable closure is the full algebraic closure if and only if
276:
1125:
85:
155:
is also the smallest algebraically closed field containing
100:. Because of this essential uniqueness, we often speak of
1049:
It can be shown along the same lines that for any subset
886:, Zorn's lemma implies that there exists a maximal ideal
123:
can be thought of as the largest algebraic extension of
337:{\displaystyle S=\{f_{\lambda }|\lambda \in \Lambda \}}
248:
of the rational numbers, e.g. the algebraic closure of
287:
Existence of an algebraic closure and splitting fields
1164:
950:
919:
851:
800:
644:
580:
554:
521:
452:
439:{\displaystyle u_{\lambda ,1},\ldots ,u_{\lambda ,d}}
394:
361:
297:
1302:. Addison-Wesley publishing Company. pp. 11–12.
1096:containing all (algebraic) separable extensions of
1212:
990:
932:
870:
825:
783:
628:{\displaystyle i\leq {\rm {degree}}(f_{\lambda })}
627:
566:
540:
499:
438:
380:
336:
222:states that the algebraic closure of the field of
271:field that contains a copy of the field of order
1501:
1368:
1372:Infinite Algebraic Extensions of Finite Fields
1220:is a non-separable algebraic field extension.
500:{\displaystyle d={\rm {degree}}(f_{\lambda })}
1116:, of degree > 1. Saying this another way,
163:is any algebraically closed field containing
147:is contained within the algebraic closure of
1323:
1213:{\displaystyle K(X)({\sqrt{X}})\supset K(X)}
331:
304:
80:, and that the algebraic closure of a field
1416:
282:(and is in fact the union of these copies).
1375:, Contemporary Mathematics, vol. 95,
1124:algebraic extension field. It is unique (
1452:
1417:Fried, Michael D.; Jarden, Moshe (2008).
1310:
1308:
1482:
913:has the property that every polynomial
14:
1502:
1305:
233:The algebraic closure of the field of
991:{\displaystyle x-(u_{\lambda ,i}+M),}
1401:
1276:
1072:
567:{\displaystyle \lambda \in \Lambda }
78:every field has an algebraic closure
1410:
1300:Introduction to commutative algebra
826:{\displaystyle r_{\lambda ,j}\in R}
24:
1046:, and hence in the union itself.
604:
601:
598:
595:
592:
589:
561:
476:
473:
470:
467:
464:
461:
328:
25:
1521:
1104:. This subextension is called a
381:{\displaystyle f_{\lambda }\in S}
185:The algebraic closure of a field
119:The algebraic closure of a field
1005:. In the same way, an extension
139:is also an algebraic closure of
135:, then the algebraic closure of
1326:Z. Math. Logik Grundlagen Math.
1485:Algebraic extensions of fields
1362:
1351:
1317:
1285:
1207:
1201:
1192:
1177:
1174:
1168:
982:
957:
871:{\displaystyle r_{\lambda ,j}}
778:
772:
704:
679:
622:
609:
541:{\displaystyle u_{\lambda ,i}}
494:
481:
318:
220:fundamental theorem of algebra
131:is any algebraic extension of
13:
1:
1377:American Mathematical Society
1269:
267:, the algebraic closure is a
178:form an algebraic closure of
1036:, and then its roots are in
933:{\displaystyle f_{\lambda }}
511:be the polynomial ring over
127:. To see this, note that if
7:
1242:
998:and hence has all roots in
212:
151:. The algebraic closure of
10:
1526:
1483:McCarthy, Paul J. (1991).
1249:Algebraically closed field
882:is strictly smaller than
388:, introduce new variables
1314:Kaplansky (1972) pp.74-76
944:splits as the product of
246:transcendental extensions
27:Algebraic field extension
1338:10.1002/malq.19920380136
1032:with sufficiently large
76:, it can be shown that
1358:Mathoverflow discussion
1231:is the Galois group of
1154:is transcendental over
349:irreducible polynomials
167:, then the elements of
1214:
992:
934:
872:
827:
785:
736:
678:
629:
568:
542:
501:
440:
382:
338:
1225:absolute Galois group
1215:
1077:An algebraic closure
993:
940:with coefficients in
935:
873:
828:
786:
710:
658:
630:
569:
543:
502:
441:
383:
339:
112:algebraic closure of
104:algebraic closure of
61:. It is one of many
1407:McCarthy (1991) p.22
1282:McCarthy (1991) p.21
1162:
948:
917:
849:
798:
642:
578:
552:
519:
450:
392:
359:
295:
201:is infinite, and is
59:algebraically closed
1254:Algebraic extension
1087:separable extension
51:algebraic extension
1379:, pp. 22–23,
1210:
1139:. For example, if
1120:is contained in a
1085:contains a unique
988:
930:
868:
823:
781:
625:
564:
538:
497:
436:
378:
344:be the set of all
334:
275:for each positive
269:countably infinite
203:countably infinite
1454:Kaplansky, Irving
1432:978-3-540-77269-9
1386:978-0-8218-5428-0
1259:Puiseux expansion
1190:
1106:separable closure
1073:Separable closure
1057:, there exists a
845:generated by the
239:algebraic numbers
74:ultrafilter lemma
40:algebraic closure
16:(Redirected from
1517:
1510:Field extensions
1496:
1479:
1458:Fields and rings
1445:
1444:
1419:Field arithmetic
1414:
1408:
1405:
1399:
1397:
1366:
1360:
1355:
1349:
1348:
1321:
1315:
1312:
1303:
1289:
1283:
1280:
1223:In general, the
1219:
1217:
1216:
1211:
1191:
1189:
1181:
1122:separably-closed
997:
995:
994:
989:
975:
974:
939:
937:
936:
931:
929:
928:
877:
875:
874:
869:
867:
866:
832:
830:
829:
824:
816:
815:
790:
788:
787:
782:
765:
764:
752:
751:
735:
724:
703:
702:
677:
672:
654:
653:
634:
632:
631:
626:
621:
620:
608:
607:
573:
571:
570:
565:
547:
545:
544:
539:
537:
536:
506:
504:
503:
498:
493:
492:
480:
479:
445:
443:
442:
437:
435:
434:
410:
409:
387:
385:
384:
379:
371:
370:
343:
341:
340:
335:
321:
316:
315:
237:is the field of
235:rational numbers
226:is the field of
96:every member of
65:in mathematics.
36:abstract algebra
21:
18:Separably closed
1525:
1524:
1520:
1519:
1518:
1516:
1515:
1514:
1500:
1499:
1468:
1449:
1448:
1433:
1423:Springer-Verlag
1415:
1411:
1406:
1402:
1387:
1367:
1363:
1356:
1352:
1322:
1318:
1313:
1306:
1296:I. G. Macdonald
1290:
1286:
1281:
1277:
1272:
1245:
1185:
1180:
1163:
1160:
1159:
1075:
1059:splitting field
1045:
1031:
1018:
1011:
1004:
964:
960:
949:
946:
945:
924:
920:
918:
915:
914:
904:
856:
852:
850:
847:
846:
805:
801:
799:
796:
795:
760:
756:
741:
737:
725:
714:
692:
688:
673:
662:
649:
645:
643:
640:
639:
616:
612:
588:
587:
579:
576:
575:
553:
550:
549:
526:
522:
520:
517:
516:
488:
484:
460:
459:
451:
448:
447:
424:
420:
399:
395:
393:
390:
389:
366:
362:
360:
357:
356:
317:
311:
307:
296:
293:
292:
289:
228:complex numbers
215:
34:, particularly
28:
23:
22:
15:
12:
11:
5:
1523:
1513:
1512:
1498:
1497:
1480:
1466:
1447:
1446:
1431:
1425:. p. 12.
1409:
1400:
1385:
1361:
1350:
1332:(4): 383–385,
1316:
1304:
1284:
1274:
1273:
1271:
1268:
1267:
1266:
1264:Complete field
1261:
1256:
1251:
1244:
1241:
1209:
1206:
1203:
1200:
1197:
1194:
1188:
1184:
1179:
1176:
1173:
1170:
1167:
1145:characteristic
1143:is a field of
1128:isomorphism).
1074:
1071:
1040:
1027:
1016:
1009:
1002:
987:
984:
981:
978:
973:
970:
967:
963:
959:
956:
953:
927:
923:
902:
894:that contains
865:
862:
859:
855:
822:
819:
814:
811:
808:
804:
792:
791:
780:
777:
774:
771:
768:
763:
759:
755:
750:
747:
744:
740:
734:
731:
728:
723:
720:
717:
713:
709:
706:
701:
698:
695:
691:
687:
684:
681:
676:
671:
668:
665:
661:
657:
652:
648:
624:
619:
615:
611:
606:
603:
600:
597:
594:
591:
586:
583:
563:
560:
557:
535:
532:
529:
525:
496:
491:
487:
483:
478:
475:
472:
469:
466:
463:
458:
455:
433:
430:
427:
423:
419:
416:
413:
408:
405:
402:
398:
377:
374:
369:
365:
333:
330:
327:
324:
320:
314:
310:
306:
303:
300:
288:
285:
284:
283:
253:
242:
231:
214:
211:
173:algebraic over
108:, rather than
72:or the weaker
26:
9:
6:
4:
3:
2:
1522:
1511:
1508:
1507:
1505:
1494:
1490:
1486:
1481:
1477:
1473:
1469:
1467:0-226-42451-0
1463:
1459:
1455:
1451:
1450:
1442:
1438:
1434:
1428:
1424:
1420:
1413:
1404:
1396:
1392:
1388:
1382:
1378:
1374:
1373:
1365:
1359:
1354:
1347:
1343:
1339:
1335:
1331:
1327:
1320:
1311:
1309:
1301:
1297:
1293:
1288:
1279:
1275:
1265:
1262:
1260:
1257:
1255:
1252:
1250:
1247:
1246:
1240:
1238:
1234:
1230:
1226:
1221:
1204:
1198:
1195:
1186:
1182:
1171:
1165:
1157:
1153:
1149:
1146:
1142:
1138:
1137:perfect field
1134:
1129:
1127:
1123:
1119:
1115:
1111:
1107:
1103:
1099:
1095:
1091:
1088:
1084:
1080:
1070:
1068:
1064:
1060:
1056:
1052:
1047:
1043:
1039:
1035:
1030:
1026:
1022:
1015:
1008:
1001:
985:
979:
976:
971:
968:
965:
961:
954:
951:
943:
925:
921:
912:
908:
901:
897:
893:
889:
885:
881:
863:
860:
857:
853:
844:
840:
836:
820:
817:
812:
809:
806:
802:
775:
769:
766:
761:
757:
753:
748:
745:
742:
738:
732:
729:
726:
721:
718:
715:
711:
707:
699:
696:
693:
689:
685:
682:
674:
669:
666:
663:
659:
655:
650:
646:
638:
637:
636:
617:
613:
584:
581:
558:
555:
533:
530:
527:
523:
515:generated by
514:
510:
489:
485:
456:
453:
431:
428:
425:
421:
417:
414:
411:
406:
403:
400:
396:
375:
372:
367:
363:
354:
350:
347:
325:
322:
312:
308:
301:
298:
281:
278:
274:
270:
266:
262:
258:
254:
251:
247:
243:
240:
236:
232:
229:
225:
221:
217:
216:
210:
208:
204:
200:
196:
192:
189:has the same
188:
183:
181:
177:
174:
170:
166:
162:
159:, because if
158:
154:
150:
146:
142:
138:
134:
130:
126:
122:
117:
115:
111:
107:
103:
99:
95:
91:
87:
83:
79:
75:
71:
66:
64:
60:
56:
52:
48:
45:
41:
37:
33:
19:
1484:
1457:
1418:
1412:
1403:
1371:
1364:
1353:
1329:
1325:
1319:
1299:
1292:M. F. Atiyah
1287:
1278:
1236:
1232:
1228:
1222:
1155:
1151:
1147:
1140:
1132:
1130:
1121:
1117:
1113:
1109:
1105:
1101:
1097:
1093:
1089:
1082:
1078:
1076:
1066:
1062:
1054:
1050:
1048:
1041:
1037:
1033:
1028:
1024:
1020:
1013:
1006:
999:
941:
910:
906:
899:
898:. The field
895:
891:
887:
883:
879:
842:
834:
793:
512:
508:
352:
290:
279:
272:
264:
263:power order
257:finite field
249:
224:real numbers
206:
198:
194:
186:
184:
179:
175:
168:
164:
160:
156:
152:
148:
144:
140:
136:
132:
128:
124:
120:
118:
113:
109:
105:
101:
97:
81:
70:Zorn's lemma
67:
54:
46:
39:
29:
355:. For each
209:is finite.
191:cardinality
90:isomorphism
32:mathematics
1493:0768.12001
1476:1001.16500
1441:1145.12001
1395:0674.12009
1346:0739.03027
1270:References
84:is unique
1196:⊃
966:λ
955:−
926:λ
858:λ
818:∈
807:λ
767:∈
754:⋅
743:λ
730:−
712:∑
694:λ
686:−
660:∏
656:−
651:λ
618:λ
585:≤
562:Λ
559:∈
556:λ
528:λ
490:λ
426:λ
415:…
401:λ
373:∈
368:λ
329:Λ
326:∈
323:λ
313:λ
171:that are
143:, and so
1504:Category
1456:(1972).
1298:(1969).
1243:See also
878:. Since
635:. Write
574:and all
548:for all
213:Examples
63:closures
57:that is
1150:and if
1100:within
837:be the
833:. Let
277:integer
1491:
1474:
1464:
1439:
1429:
1393:
1383:
1344:
507:. Let
446:where
255:For a
68:Using
49:is an
1235:over
1135:is a
1126:up to
1065:over
839:ideal
794:with
346:monic
261:prime
94:fixes
92:that
86:up to
44:field
42:of a
38:, an
1462:ISBN
1427:ISBN
1381:ISBN
1294:and
291:Let
252:(π).
218:The
1489:Zbl
1472:Zbl
1437:Zbl
1391:Zbl
1342:Zbl
1334:doi
1227:of
1108:of
1092:of
1081:of
1061:of
1053:of
1012:of
890:in
841:in
351:in
259:of
205:if
197:if
193:as
102:the
88:an
53:of
30:In
1506::
1470:.
1435:.
1389:,
1340:,
1330:38
1328:,
1307:^
1239:.
1158:,
1069:.
1044:+1
182:.
116:.
110:an
1495:.
1478:.
1443:.
1398:.
1336::
1237:K
1233:K
1229:K
1208:)
1205:X
1202:(
1199:K
1193:)
1187:p
1183:X
1178:(
1175:)
1172:X
1169:(
1166:K
1156:K
1152:X
1148:p
1141:K
1133:K
1118:K
1114:K
1110:K
1102:K
1098:K
1094:K
1090:K
1083:K
1079:K
1067:K
1063:S
1055:K
1051:S
1042:n
1038:K
1034:n
1029:n
1025:K
1021:K
1017:1
1014:K
1010:2
1007:K
1003:1
1000:K
986:,
983:)
980:M
977:+
972:i
969:,
962:u
958:(
952:x
942:K
922:f
911:M
909:/
907:R
905:=
903:1
900:K
896:I
892:R
888:M
884:R
880:I
864:j
861:,
854:r
843:R
835:I
821:R
813:j
810:,
803:r
779:]
776:x
773:[
770:R
762:j
758:x
749:j
746:,
739:r
733:1
727:d
722:0
719:=
716:j
708:=
705:)
700:i
697:,
690:u
683:x
680:(
675:d
670:1
667:=
664:i
647:f
623:)
614:f
610:(
605:e
602:e
599:r
596:g
593:e
590:d
582:i
534:i
531:,
524:u
513:K
509:R
495:)
486:f
482:(
477:e
474:e
471:r
468:g
465:e
462:d
457:=
454:d
432:d
429:,
422:u
418:,
412:,
407:1
404:,
397:u
376:S
364:f
353:K
332:}
319:|
309:f
305:{
302:=
299:S
280:n
273:q
265:q
250:Q
241:.
230:.
207:K
199:K
195:K
187:K
180:K
176:K
169:M
165:K
161:M
157:K
153:K
149:K
145:L
141:K
137:L
133:K
129:L
125:K
121:K
114:K
106:K
98:K
82:K
55:K
47:K
20:)
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