36:
965:
1270:
1330:
1035:
1401:
1357:
1545:
1589:
1716:
761:
658:
476:
1430:
839:
691:
558:
509:
1923:
1903:
1883:
1856:
1836:
1816:
1796:
1776:
1756:
1736:
1672:
1652:
1632:
1612:
1487:
1463:
1200:
1180:
1156:
1132:
1112:
1088:
1055:
879:
859:
810:
790:
711:
618:
598:
578:
529:
436:
416:
393:
373:
1925:
is algebraically closed, which can again be directly shown using the characterisation of simple algebraic extensions above. An example for this is the
65:
891:
1205:
17:
1967:
1466:
1279:
1159:
87:
1594:
This characterization can be used to show that the sum, difference, product and quotient of algebraic elements over
58:
1778:
also yields a finite extension, and therefore these elements are algebraic as well. Thus set of all elements of
1489:, by the universal property of the field of fractions. We can conclude that in any case, we find an isomorphism
1959:
1273:
978:
1362:
661:
322:) are both real, but not of any polynomial with only rational coefficients. (The definition of the term
1862:
886:
1335:
2001:
1492:
48:
1550:
52:
44:
1861:
Fields that do not allow any algebraic elements over them (except their own elements) are called
1938:
1677:
1135:
69:
323:
215:
722:
626:
444:
1977:
1985:
1406:
815:
667:
534:
485:
8:
1058:
1908:
1888:
1868:
1841:
1821:
1801:
1781:
1761:
1741:
1721:
1657:
1637:
1617:
1597:
1472:
1439:
1433:
1185:
1165:
1141:
1117:
1097:
1064:
1040:
864:
844:
795:
766:
696:
603:
583:
563:
514:
421:
401:
378:
358:
1963:
1981:
1926:
1091:
258:
211:
972:
1973:
245:
111:
235:
1995:
968:
714:
295:
166:
101:
1951:
152:
1962:, vol. 211 (Revised third ed.), New York: Springer-Verlag,
960:{\displaystyle \varepsilon _{a}:K\rightarrow K(a),\,P\mapsto P(a)}
1885:
is algebraically closed, then the field of algebraic elements of
1265:{\displaystyle K/(p)\rightarrow \mathrm {im} (\varepsilon _{a})}
1591:. Investigating this construction yields the desired results.
1718:
is finite. As it contains the aforementioned combinations of
1276:
is an isomorphism of fields, where we can then observe that
355:
The following conditions are equivalent for an element
285:
1911:
1891:
1871:
1844:
1824:
1804:
1784:
1764:
1744:
1724:
1680:
1660:
1640:
1620:
1600:
1553:
1495:
1475:
1442:
1409:
1365:
1359:
is injective and hence we obtain a field isomorphism
1338:
1282:
1208:
1188:
1168:
1144:
1120:
1100:
1067:
1043:
981:
894:
867:
847:
818:
798:
769:
725:
699:
670:
629:
606:
586:
566:
537:
517:
488:
447:
424:
404:
381:
361:
1325:{\displaystyle \mathrm {im} (\varepsilon _{a})=K(a)}
1917:
1897:
1877:
1850:
1830:
1810:
1790:
1770:
1750:
1730:
1710:
1666:
1646:
1626:
1606:
1583:
1539:
1481:
1457:
1424:
1395:
1351:
1324:
1264:
1194:
1174:
1150:
1134:, which then also generates the ideal and must be
1126:
1106:
1082:
1049:
1029:
959:
873:
853:
833:
804:
784:
755:
705:
685:
652:
612:
592:
572:
552:
523:
503:
470:
430:
410:
387:
367:
1865:. The field of complex numbers is an example. If
1993:
57:but its sources remain unclear because it lacks
1182:and it encodes many important properties of
1114:with minimal degree and leading coefficient
1024:
982:
938:
885:To make this more explicit, consider the
267:, since it is the root of the polynomial
88:Learn how and when to remove this message
14:
1994:
1061:contains non-zero polynomials, but as
1030:{\displaystyle \{P\in K\mid P(a)=0\}}
1950:
1396:{\displaystyle K(X)\rightarrow K(a)}
29:
24:
1287:
1284:
1242:
1239:
1094:, it contains a unique polynomial
25:
2013:
560:denotes the smallest subfield of
1818:is a field that sits in between
1352:{\displaystyle \varepsilon _{a}}
812:that can be written in the form
294:but algebraic over the field of
282:whose coefficients are rational.
151:, if there exists some non-zero
34:
1540:{\displaystyle K(a)\cong K/(p)}
1705:
1699:
1696:
1693:
1687:
1681:
1584:{\displaystyle K(a)\cong K(X)}
1578:
1572:
1563:
1557:
1534:
1528:
1520:
1514:
1505:
1499:
1452:
1446:
1419:
1413:
1390:
1384:
1378:
1375:
1369:
1319:
1313:
1304:
1291:
1259:
1246:
1235:
1232:
1226:
1218:
1212:
1077:
1071:
1015:
1009:
1000:
994:
954:
948:
942:
932:
926:
920:
917:
911:
828:
822:
792:is the set of all elements of
779:
773:
750:
744:
735:
729:
680:
674:
639:
633:
547:
541:
498:
492:
457:
451:
218:(where the field extension is
13:
1:
1960:Graduate Texts in Mathematics
1944:
1202:. Hence the ring isomorphism
350:
318:, whose coefficients (1 and −
210:These notions generalize the
660:has finite degree, i.e. the
192:that are not algebraic over
7:
1932:
1758:, adjoining one of them to
1467:field of rational functions
251:
27:Concept in abstract algebra
10:
2018:
1927:field of algebraic numbers
861:whose coefficients lie in
1711:{\displaystyle (K(a))(b)}
1674:are both algebraic, then
1614:are again algebraic over
1798:that are algebraic over
43:This article includes a
288:is transcendental over
72:more precise citations.
1939:Algebraic independence
1919:
1899:
1879:
1852:
1832:
1812:
1792:
1772:
1752:
1732:
1712:
1668:
1648:
1628:
1608:
1585:
1541:
1483:
1459:
1426:
1397:
1353:
1326:
1266:
1196:
1176:
1152:
1128:
1108:
1084:
1051:
1031:
961:
875:
855:
835:
806:
786:
757:
756:{\displaystyle K=K(a)}
707:
687:
654:
653:{\displaystyle K(a)/K}
614:
594:
574:
554:
525:
505:
472:
471:{\displaystyle K(a)/K}
432:
412:
389:
369:
216:transcendental numbers
18:Transcendental element
1920:
1900:
1880:
1853:
1833:
1813:
1793:
1773:
1753:
1733:
1713:
1669:
1649:
1629:
1609:
1586:
1542:
1484:
1460:
1427:
1398:
1354:
1327:
1267:
1197:
1177:
1153:
1129:
1109:
1085:
1052:
1032:
962:
887:polynomial evaluation
876:
856:
836:
807:
787:
758:
708:
688:
655:
615:
595:
575:
555:
526:
506:
473:
433:
413:
390:
370:
324:transcendental number
1909:
1889:
1869:
1863:algebraically closed
1842:
1822:
1802:
1782:
1762:
1742:
1722:
1678:
1658:
1638:
1618:
1598:
1551:
1493:
1473:
1440:
1425:{\displaystyle K(X)}
1407:
1363:
1336:
1280:
1274:homomorphism theorem
1206:
1186:
1166:
1142:
1118:
1098:
1065:
1041:
979:
892:
865:
845:
834:{\displaystyle g(a)}
816:
796:
767:
723:
697:
686:{\displaystyle K(a)}
668:
627:
623:the field extension
604:
584:
564:
553:{\displaystyle K(a)}
535:
515:
504:{\displaystyle K(a)}
486:
445:
441:the field extension
422:
402:
379:
359:
303:: it is the root of
1057:is algebraic, this
478:is algebraic, i.e.
244:being the field of
234:being the field of
200:transcendental over
1915:
1895:
1875:
1848:
1828:
1808:
1788:
1768:
1748:
1728:
1708:
1664:
1644:
1624:
1604:
1581:
1537:
1479:
1455:
1434:field of fractions
1422:
1393:
1349:
1322:
1262:
1192:
1172:
1160:minimal polynomial
1148:
1124:
1104:
1080:
1047:
1027:
957:
871:
851:
841:with a polynomial
831:
802:
782:
753:
703:
683:
650:
610:
590:
570:
550:
521:
511:is algebraic over
501:
468:
428:
418:is algebraic over
408:
385:
365:
261:is algebraic over
120:, then an element
45:list of references
1969:978-0-387-95385-4
1918:{\displaystyle K}
1898:{\displaystyle L}
1878:{\displaystyle L}
1851:{\displaystyle K}
1831:{\displaystyle L}
1811:{\displaystyle K}
1791:{\displaystyle L}
1771:{\displaystyle K}
1751:{\displaystyle b}
1731:{\displaystyle a}
1667:{\displaystyle b}
1647:{\displaystyle a}
1627:{\displaystyle K}
1607:{\displaystyle K}
1482:{\displaystyle K}
1458:{\displaystyle K}
1195:{\displaystyle a}
1175:{\displaystyle a}
1151:{\displaystyle p}
1138:. The polynomial
1127:{\displaystyle 1}
1107:{\displaystyle p}
1083:{\displaystyle K}
1050:{\displaystyle a}
874:{\displaystyle K}
854:{\displaystyle g}
805:{\displaystyle L}
785:{\displaystyle K}
706:{\displaystyle K}
613:{\displaystyle a}
593:{\displaystyle K}
573:{\displaystyle L}
524:{\displaystyle K}
431:{\displaystyle K}
411:{\displaystyle a}
388:{\displaystyle L}
368:{\displaystyle a}
212:algebraic numbers
134:algebraic element
98:
97:
90:
16:(Redirected from
2009:
2002:Abstract algebra
1988:
1924:
1922:
1921:
1916:
1904:
1902:
1901:
1896:
1884:
1882:
1881:
1876:
1857:
1855:
1854:
1849:
1837:
1835:
1834:
1829:
1817:
1815:
1814:
1809:
1797:
1795:
1794:
1789:
1777:
1775:
1774:
1769:
1757:
1755:
1754:
1749:
1737:
1735:
1734:
1729:
1717:
1715:
1714:
1709:
1673:
1671:
1670:
1665:
1653:
1651:
1650:
1645:
1633:
1631:
1630:
1625:
1613:
1611:
1610:
1605:
1590:
1588:
1587:
1582:
1546:
1544:
1543:
1538:
1527:
1488:
1486:
1485:
1480:
1464:
1462:
1461:
1456:
1431:
1429:
1428:
1423:
1402:
1400:
1399:
1394:
1358:
1356:
1355:
1350:
1348:
1347:
1331:
1329:
1328:
1323:
1303:
1302:
1290:
1272:obtained by the
1271:
1269:
1268:
1263:
1258:
1257:
1245:
1225:
1201:
1199:
1198:
1193:
1181:
1179:
1178:
1173:
1157:
1155:
1154:
1149:
1133:
1131:
1130:
1125:
1113:
1111:
1110:
1105:
1092:euclidean domain
1089:
1087:
1086:
1081:
1056:
1054:
1053:
1048:
1036:
1034:
1033:
1028:
966:
964:
963:
958:
904:
903:
880:
878:
877:
872:
860:
858:
857:
852:
840:
838:
837:
832:
811:
809:
808:
803:
791:
789:
788:
783:
762:
760:
759:
754:
712:
710:
709:
704:
692:
690:
689:
684:
659:
657:
656:
651:
646:
619:
617:
616:
611:
599:
597:
596:
591:
579:
577:
576:
571:
559:
557:
556:
551:
530:
528:
527:
522:
510:
508:
507:
502:
477:
475:
474:
469:
464:
437:
435:
434:
429:
417:
415:
414:
409:
394:
392:
391:
386:
374:
372:
371:
366:
345:
335:
321:
317:
302:
293:
281:
266:
259:square root of 2
246:rational numbers
243:
233:
227:
206:
197:
191:
185:
174:
164:
150:
141:
131:
125:
119:
109:
93:
86:
82:
79:
73:
68:this article by
59:inline citations
38:
37:
30:
21:
2017:
2016:
2012:
2011:
2010:
2008:
2007:
2006:
1992:
1991:
1970:
1947:
1935:
1910:
1907:
1906:
1890:
1887:
1886:
1870:
1867:
1866:
1843:
1840:
1839:
1823:
1820:
1819:
1803:
1800:
1799:
1783:
1780:
1779:
1763:
1760:
1759:
1743:
1740:
1739:
1723:
1720:
1719:
1679:
1676:
1675:
1659:
1656:
1655:
1639:
1636:
1635:
1619:
1616:
1615:
1599:
1596:
1595:
1552:
1549:
1548:
1523:
1494:
1491:
1490:
1474:
1471:
1470:
1441:
1438:
1437:
1408:
1405:
1404:
1364:
1361:
1360:
1343:
1339:
1337:
1334:
1333:
1298:
1294:
1283:
1281:
1278:
1277:
1253:
1249:
1238:
1221:
1207:
1204:
1203:
1187:
1184:
1183:
1167:
1164:
1163:
1143:
1140:
1139:
1119:
1116:
1115:
1099:
1096:
1095:
1066:
1063:
1062:
1042:
1039:
1038:
980:
977:
976:
899:
895:
893:
890:
889:
866:
863:
862:
846:
843:
842:
817:
814:
813:
797:
794:
793:
768:
765:
764:
724:
721:
720:
698:
695:
694:
669:
666:
665:
642:
628:
625:
624:
605:
602:
601:
585:
582:
581:
565:
562:
561:
536:
533:
532:
516:
513:
512:
487:
484:
483:
460:
446:
443:
442:
423:
420:
419:
403:
400:
399:
380:
377:
376:
360:
357:
356:
353:
337:
327:
319:
304:
298:
289:
268:
262:
254:
239:
236:complex numbers
229:
219:
202:
193:
187:
176:
170:
155:
146:
137:
127:
121:
115:
112:extension field
105:
94:
83:
77:
74:
63:
49:related reading
39:
35:
28:
23:
22:
15:
12:
11:
5:
2015:
2005:
2004:
1990:
1989:
1968:
1946:
1943:
1942:
1941:
1934:
1931:
1914:
1894:
1874:
1847:
1827:
1807:
1787:
1767:
1747:
1727:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1686:
1683:
1663:
1643:
1623:
1603:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1556:
1536:
1533:
1530:
1526:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1478:
1454:
1451:
1448:
1445:
1421:
1418:
1415:
1412:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1346:
1342:
1321:
1318:
1315:
1312:
1309:
1306:
1301:
1297:
1293:
1289:
1286:
1261:
1256:
1252:
1248:
1244:
1241:
1237:
1234:
1231:
1228:
1224:
1220:
1217:
1214:
1211:
1191:
1171:
1158:is called the
1147:
1123:
1103:
1079:
1076:
1073:
1070:
1046:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
956:
953:
950:
947:
944:
941:
937:
934:
931:
928:
925:
922:
919:
916:
913:
910:
907:
902:
898:
883:
882:
870:
850:
830:
827:
824:
821:
801:
781:
778:
775:
772:
752:
749:
746:
743:
740:
737:
734:
731:
728:
718:
702:
682:
679:
676:
673:
649:
645:
641:
638:
635:
632:
621:
609:
589:
569:
549:
546:
543:
540:
520:
500:
497:
494:
491:
467:
463:
459:
456:
453:
450:
439:
427:
407:
384:
364:
352:
349:
348:
347:
283:
253:
250:
186:. Elements of
144:algebraic over
96:
95:
53:external links
42:
40:
33:
26:
9:
6:
4:
3:
2:
2014:
2003:
2000:
1999:
1997:
1987:
1983:
1979:
1975:
1971:
1965:
1961:
1957:
1953:
1949:
1948:
1940:
1937:
1936:
1930:
1928:
1912:
1892:
1872:
1864:
1859:
1845:
1825:
1805:
1785:
1765:
1745:
1725:
1702:
1690:
1684:
1661:
1641:
1621:
1601:
1592:
1575:
1569:
1566:
1560:
1554:
1531:
1524:
1517:
1511:
1508:
1502:
1496:
1476:
1468:
1449:
1443:
1435:
1416:
1410:
1387:
1381:
1372:
1366:
1344:
1340:
1332:. Otherwise,
1316:
1310:
1307:
1299:
1295:
1275:
1254:
1250:
1229:
1222:
1215:
1209:
1189:
1169:
1161:
1145:
1137:
1121:
1101:
1093:
1074:
1068:
1060:
1044:
1021:
1018:
1012:
1006:
1003:
997:
991:
988:
985:
974:
970:
951:
945:
939:
935:
929:
923:
914:
908:
905:
900:
896:
888:
868:
848:
825:
819:
799:
776:
770:
747:
741:
738:
732:
726:
719:
716:
700:
677:
671:
663:
647:
643:
636:
630:
622:
607:
587:
567:
544:
538:
518:
495:
489:
481:
465:
461:
454:
448:
440:
425:
405:
398:
397:
396:
382:
362:
344:
340:
334:
330:
325:
315:
311:
307:
301:
297:
292:
287:
284:
279:
275:
271:
265:
260:
256:
255:
249:
247:
242:
237:
232:
226:
222:
217:
213:
208:
205:
201:
196:
190:
183:
179:
173:
168:
162:
158:
154:
149:
145:
140:
135:
132:is called an
130:
124:
118:
113:
108:
103:
92:
89:
81:
71:
67:
61:
60:
54:
50:
46:
41:
32:
31:
19:
1955:
1860:
1593:
969:homomorphism
967:. This is a
884:
715:vector space
479:
354:
342:
338:
332:
328:
313:
309:
305:
299:
296:real numbers
290:
277:
273:
269:
263:
240:
230:
224:
220:
209:
203:
199:
194:
188:
181:
177:
171:
167:coefficients
160:
156:
147:
143:
138:
133:
128:
122:
116:
106:
99:
84:
75:
64:Please help
56:
1952:Lang, Serge
1465:, i.e. the
1136:irreducible
580:containing
482:element of
198:are called
102:mathematics
70:introducing
1986:0984.00001
1945:References
717:is finite,
351:Properties
175:such that
153:polynomial
142:, or just
78:March 2013
1634:. For if
1567:≅
1509:≅
1379:→
1341:ε
1296:ε
1251:ε
1236:→
1004:∣
989:∈
943:↦
921:→
897:ε
662:dimension
1996:Category
1954:(2002),
1933:See also
1403:, where
971:and its
763:, where
252:Examples
214:and the
1978:1878556
1956:Algebra
1432:is the
228:, with
66:improve
1984:
1976:
1966:
973:kernel
531:(here
336:, not
110:is an
1905:over
1090:is a
1059:ideal
1037:. If
693:as a
480:every
326:uses
184:) = 0
165:with
136:over
104:, if
51:, or
1964:ISBN
1838:and
1738:and
1654:and
600:and
312:) =
276:) =
257:The
238:and
1982:Zbl
1858:.
1547:or
1469:on
1436:of
1162:of
975:is
664:of
375:of
316:− π
280:− 2
248:).
169:in
126:of
114:of
100:In
1998::
1980:,
1974:MR
1972:,
1958:,
1929:.
620:),
395::
346:.)
286:Pi
207:.
55:,
47:,
1913:K
1893:L
1873:L
1846:K
1826:L
1806:K
1786:L
1766:K
1746:b
1726:a
1706:)
1703:b
1700:(
1697:)
1694:)
1691:a
1688:(
1685:K
1682:(
1662:b
1642:a
1622:K
1602:K
1579:)
1576:X
1573:(
1570:K
1564:)
1561:a
1558:(
1555:K
1535:)
1532:p
1529:(
1525:/
1521:]
1518:X
1515:[
1512:K
1506:)
1503:a
1500:(
1497:K
1477:K
1453:]
1450:X
1447:[
1444:K
1420:)
1417:X
1414:(
1411:K
1391:)
1388:a
1385:(
1382:K
1376:)
1373:X
1370:(
1367:K
1345:a
1320:)
1317:a
1314:(
1311:K
1308:=
1305:)
1300:a
1292:(
1288:m
1285:i
1260:)
1255:a
1247:(
1243:m
1240:i
1233:)
1230:p
1227:(
1223:/
1219:]
1216:X
1213:[
1210:K
1190:a
1170:a
1146:p
1122:1
1102:p
1078:]
1075:X
1072:[
1069:K
1045:a
1025:}
1022:0
1019:=
1016:)
1013:a
1010:(
1007:P
1001:]
998:X
995:[
992:K
986:P
983:{
955:)
952:a
949:(
946:P
940:P
936:,
933:)
930:a
927:(
924:K
918:]
915:X
912:[
909:K
906::
901:a
881:.
869:K
849:g
829:)
826:a
823:(
820:g
800:L
780:]
777:a
774:[
771:K
751:)
748:a
745:(
742:K
739:=
736:]
733:a
730:[
727:K
713:-
701:K
681:)
678:a
675:(
672:K
648:K
644:/
640:)
637:a
634:(
631:K
608:a
588:K
568:L
548:)
545:a
542:(
539:K
519:K
499:)
496:a
493:(
490:K
466:K
462:/
458:)
455:a
452:(
449:K
438:,
426:K
406:a
383:L
363:a
343:R
341:/
339:C
333:Q
331:/
329:C
320:π
314:x
310:x
308:(
306:g
300:R
291:Q
278:x
274:x
272:(
270:g
264:Q
241:Q
231:C
225:Q
223:/
221:C
204:K
195:K
189:L
182:a
180:(
178:g
172:K
163:)
161:x
159:(
157:g
148:K
139:K
129:L
123:a
117:K
107:L
91:)
85:(
80:)
76:(
62:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.