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