238:
137:
160:
The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately.
1574:
1310:
of prime order. Two prime fields of the same characteristic are isomorphic, and this isomorphism is unique. In other words, there is essentially a unique prime field in each characteristic.
1628:
1521:
1482:
1051:
976:
697:
417:
1738:
858:
1638:
we know that the sizes of finite vector spaces over finite fields are a power of the size of the field. This also shows that the size of any finite vector space is a prime power.
1016:
1308:
790:
1678:
1415:
1385:
1360:
1279:
934:
912:
886:
348:
317:
1654:
The requirements of ring homomorphisms are such that there can be only one (in fact, exactly one) homomorphism from the ring of integers to any ring; in the language of
1330:
are characteristic zero fields that are widely used in number theory. They have absolute values which are very different from those of complex numbers.
261:
184:
83:
1805:
615:. This can sometimes be used to exclude the possibility of certain ring homomorphisms. The only ring with characteristic
1529:
1688:. Again this applies when a ring has a multiplicative identity element (which is preserved by ring homomorphisms).
1598:
1491:
1452:
1021:
946:
667:
387:
802:
1830:
993:
1284:
766:
1661:
1398:
1368:
1343:
1262:
917:
895:
869:
331:
300:
165:
1762:
1392:
1180:
760:
476:
431:
1825:
321:
8:
1524:
1146:
646:
420:
1797:
700:
587:
28:
1801:
1744:
1685:
1485:
1446:
1166:
591:
378:
325:
53:
1770:
1758:
1445:
There exist infinite fields of prime characteristic. For example, the field of all
266:
45:
1655:
1338:
1256:
1244:
1191:
650:
1681:
1635:
1323:
889:
360:
285:
169:
1774:
255:
exists; otherwise zero). This definition applies in the more general class of
1819:
1334:
797:
654:
256:
1697:
It is a vector space over a finite field, which we have shown to be of size
442:
is the largest. Then the characteristic of a ring is the smallest value of
60:). If no such number exists, the ring is said to have characteristic zero.
1631:
1426:
1327:
1210:
or a prime number. A field of non-zero characteristic is called a field of
642:
634:
1580:
1363:
463:. This is the appropriate partial ordering because of such facts as that
382:
20:
44:, is defined to be the smallest positive number of copies of the ring's
265:); for (unital) rings the two definitions are equivalent due to their
1195:
620:
262:
Ring (mathematics) § Multiplicative identity and the term "ring"
233:{\displaystyle \underbrace {a+\cdots +a} _{n{\text{ summands}}}=0}
132:{\displaystyle \underbrace {1+\cdots +1} _{n{\text{ summands}}}=0}
721:
375:
1206:
As mentioned above, the characteristic of any field is either
978:-algebra is equivalently a ring whose characteristic divides
1234:; otherwise it has the same value as the characteristic.
1763:"5. Characteristic exponent of a field. Perfect fields"
1732:
1730:
1664:
1601:
1532:
1494:
1455:
1401:
1371:
1346:
1287:
1265:
1024:
996:
949:
920:
898:
872:
805:
769:
670:
390:
334:
303:
187:
86:
1727:
1672:
1622:
1568:
1515:
1476:
1409:
1379:
1354:
1302:
1273:
1045:
1010:
970:
928:
906:
880:
852:
784:
691:
411:
342:
311:
232:
131:
1420:
1226:is defined similarly, except that it is equal to
16:Smallest integer n for which n equals 0 in a ring
1817:
1313:
1569:{\displaystyle \mathbb {Z} /p\mathbb {Z} ((T))}
455:. If nothing "smaller" (in this ordering) than
272:
164:The characteristic may also be taken to be the
742:have the same characteristic. For example, if
459:will suffice, then the characteristic is
1736:
1255:. This subfield is isomorphic to either the
1737:Fraleigh, John B.; Brand, Neal E. (2020).
1623:{\displaystyle \mathbb {Z} /p\mathbb {Z} }
1516:{\displaystyle \mathbb {Z} /p\mathbb {Z} }
1477:{\displaystyle \mathbb {Z} /p\mathbb {Z} }
1046:{\displaystyle \mathbb {Z} /n\mathbb {Z} }
971:{\displaystyle \mathbb {Z} /n\mathbb {Z} }
692:{\displaystyle \mathbb {Z} /n\mathbb {Z} }
412:{\displaystyle \mathbb {Z} /n\mathbb {Z} }
1666:
1616:
1603:
1547:
1534:
1509:
1496:
1470:
1457:
1403:
1373:
1348:
1290:
1267:
1039:
1026:
998:
964:
951:
922:
900:
874:
808:
772:
685:
672:
405:
392:
336:
305:
172:, that is, the smallest positive integer
1757:
853:{\displaystyle \mathbb {F} _{p}/(q(X))}
1818:
1791:
1053:if and only if the characteristic of
645:. In particular, this applies to all
637:, then its characteristic is either
13:
1740:A First Course in Abstract Algebra
623:, which has only a single element
14:
1842:
1595:. Since in that case it contains
1395:and the field of complex numbers
1201:
1011:{\displaystyle \mathbb {Z} \to R}
984:. This is because for every ring
1303:{\displaystyle \mathbb {F} _{p}}
785:{\displaystyle \mathbb {F} _{p}}
569:
493:, and that no ring homomorphism
71:is the smallest positive number
1018:, and this map factors through
763:with coefficients in the field
426:When the non-negative integers
1751:
1691:
1648:
1563:
1560:
1554:
1551:
1421:Fields of prime characteristic
1002:
847:
844:
838:
832:
824:
818:
609:divides the characteristic of
1:
1721:
1314:Fields of characteristic zero
990:there is a ring homomorphism
866:. Another example: The field
860:is a field of characteristic
657:. Any ring of characteristic
633:does not have any nontrivial
603:, then the characteristic of
524:The characteristic of a ring
278:The characteristic of a ring
155:
1673:{\displaystyle \mathbb {Z} }
1417:are of characteristic zero.
1410:{\displaystyle \mathbb {C} }
1380:{\displaystyle \mathbb {R} }
1355:{\displaystyle \mathbb {Q} }
1274:{\displaystyle \mathbb {Q} }
929:{\displaystyle \mathbb {C} }
907:{\displaystyle \mathbb {Z} }
881:{\displaystyle \mathbb {C} }
343:{\displaystyle \mathbb {Z} }
312:{\displaystyle \mathbb {Z} }
273:Equivalent characterizations
7:
1769:. Springer. p. A.V.7.
1230:when the characteristic is
914:, so the characteristic of
536:precisely if the statement
10:
1847:
1784:
1634:over that field, and from
1318:The most common fields of
1145:– the normally incorrect "
423:of the above homomorphism.
359:The characteristic is the
1775:10.1007/978-3-642-61698-3
1322:are the subfields of the
1071:in the ring, then adding
1792:McCoy, Neal H. (1973) .
1767:Algebra II, Chapters 4–7
1641:
1583:of prime characteristic
1387:, the characteristic is
1337:, such as the field of
1224:characteristic exponent
1216:positive characteristic
1065:. In this case for any
627:. If a nontrivial ring
249:of the ring (again, if
52:) that will sum to the
46:multiplicative identity
1674:
1624:
1570:
1517:
1478:
1411:
1393:algebraic number field
1381:
1356:
1304:
1275:
1181:Frobenius homomorphism
1178:, which is called the
1093:If a commutative ring
1047:
1012:
972:
930:
908:
882:
854:
786:
761:irreducible polynomial
693:
434:by divisibility, then
413:
344:
313:
234:
133:
1675:
1625:
1571:
1525:formal Laurent series
1518:
1479:
1412:
1382:
1357:
1305:
1276:
1243:has a unique minimal
1212:finite characteristic
1048:
1013:
973:
931:
909:
883:
855:
787:
694:
477:least common multiple
414:
345:
314:
235:
134:
1662:
1599:
1530:
1492:
1453:
1399:
1369:
1344:
1285:
1263:
1220:prime characteristic
1101:prime characteristic
1022:
994:
947:
918:
896:
870:
803:
767:
668:
438:is the smallest and
388:
332:
301:
185:
84:
1831:Field (mathematics)
1794:The Theory of Rings
1436:has characteristic
1320:characteristic zero
796:elements, then the
708:has characteristic
590:and there exists a
1798:Chelsea Publishing
1670:
1620:
1566:
1513:
1474:
1447:rational functions
1407:
1377:
1352:
1300:
1281:or a finite field
1271:
1247:, also called its
1149:" holds for power
1043:
1008:
968:
926:
904:
878:
850:
782:
689:
409:
340:
309:
243:for every element
230:
223:
211:
129:
122:
110:
1807:978-0-8284-0266-8
1759:Bourbaki, Nicolas
1745:Pearson Education
1703:, so its size is
1686:category of rings
1486:algebraic closure
1167:ring homomorphism
1127:for all elements
592:ring homomorphism
559:is a multiple of
432:partially ordered
428:{0, 1, 2, 3, ...}
326:ring homomorphism
220:
190:
188:
142:if such a number
119:
89:
87:
54:additive identity
1838:
1811:
1779:
1778:
1755:
1749:
1748:
1743:(8th ed.).
1734:
1715:
1713:
1702:
1695:
1689:
1679:
1677:
1676:
1671:
1669:
1652:
1629:
1627:
1626:
1621:
1619:
1611:
1606:
1594:
1588:
1579:The size of any
1575:
1573:
1572:
1567:
1550:
1542:
1537:
1523:or the field of
1522:
1520:
1519:
1514:
1512:
1504:
1499:
1483:
1481:
1480:
1475:
1473:
1465:
1460:
1441:
1435:
1416:
1414:
1413:
1408:
1406:
1390:
1386:
1384:
1383:
1378:
1376:
1362:or the field of
1361:
1359:
1358:
1353:
1351:
1339:rational numbers
1309:
1307:
1306:
1301:
1299:
1298:
1293:
1280:
1278:
1277:
1272:
1270:
1253:
1252:
1242:
1233:
1229:
1209:
1189:
1177:
1164:
1154:
1147:freshman's dream
1144:
1138:
1132:
1126:
1107:
1098:
1089:
1082:
1076:
1070:
1064:
1058:
1052:
1050:
1049:
1044:
1042:
1034:
1029:
1017:
1015:
1014:
1009:
1001:
989:
983:
977:
975:
974:
969:
967:
959:
954:
939:
935:
933:
932:
927:
925:
913:
911:
910:
905:
903:
887:
885:
884:
879:
877:
865:
859:
857:
856:
851:
831:
817:
816:
811:
795:
791:
789:
788:
783:
781:
780:
775:
758:
747:
741:
735:
729:
719:
713:
707:
698:
696:
695:
690:
688:
680:
675:
660:
651:integral domains
640:
632:
626:
618:
614:
608:
602:
585:
579:
564:
558:
552:
542:
535:
529:
520:
513:
506:
492:
485:
474:
462:
458:
454:
447:
441:
437:
429:
418:
416:
415:
410:
408:
400:
395:
373:
367:
355:
349:
347:
346:
341:
339:
319:
318:
316:
315:
310:
308:
292:
283:
267:distributive law
254:
248:
239:
237:
236:
231:
222:
221:
218:
212:
207:
177:
151:
147:
138:
136:
135:
130:
121:
120:
117:
111:
106:
76:
70:
59:
51:
43:
36:, often denoted
35:
1846:
1845:
1841:
1840:
1839:
1837:
1836:
1835:
1816:
1815:
1814:
1808:
1787:
1782:
1756:
1752:
1735:
1728:
1724:
1719:
1718:
1704:
1698:
1696:
1692:
1665:
1663:
1660:
1659:
1656:category theory
1653:
1649:
1644:
1615:
1607:
1602:
1600:
1597:
1596:
1590:
1584:
1546:
1538:
1533:
1531:
1528:
1527:
1508:
1500:
1495:
1493:
1490:
1489:
1469:
1461:
1456:
1454:
1451:
1450:
1437:
1429:
1423:
1402:
1400:
1397:
1396:
1388:
1372:
1370:
1367:
1366:
1347:
1345:
1342:
1341:
1324:complex numbers
1316:
1294:
1289:
1288:
1286:
1283:
1282:
1266:
1264:
1261:
1260:
1257:rational number
1250:
1249:
1238:
1231:
1227:
1207:
1204:
1202:Case of fields
1192:integral domain
1185:
1169:
1165:then defines a
1156:
1150:
1140:
1134:
1128:
1109:
1108:, then we have
1103:
1094:
1084:
1078:
1072:
1066:
1060:
1054:
1038:
1030:
1025:
1023:
1020:
1019:
997:
995:
992:
991:
985:
979:
963:
955:
950:
948:
945:
944:
937:
921:
919:
916:
915:
899:
897:
894:
893:
890:complex numbers
873:
871:
868:
867:
861:
827:
812:
807:
806:
804:
801:
800:
793:
776:
771:
770:
768:
765:
764:
749:
743:
737:
731:
725:
715:
709:
703:
684:
676:
671:
669:
666:
665:
658:
638:
628:
624:
616:
610:
604:
594:
581:
575:
572:
560:
554:
544:
537:
531:
525:
515:
508:
494:
487:
480:
464:
460:
456:
449:
443:
439:
435:
427:
419:, which is the
404:
396:
391:
389:
386:
385:
369:
363:
351:
335:
333:
330:
329:
304:
302:
299:
298:
294:
288:
279:
275:
250:
244:
217:
213:
191:
189:
186:
183:
182:
173:
158:
149:
143:
116:
112:
90:
88:
85:
82:
81:
72:
64:
57:
49:
37:
31:
17:
12:
11:
5:
1844:
1834:
1833:
1828:
1813:
1812:
1806:
1788:
1786:
1783:
1781:
1780:
1750:
1725:
1723:
1720:
1717:
1716:
1690:
1682:initial object
1668:
1646:
1645:
1643:
1640:
1636:linear algebra
1618:
1614:
1610:
1605:
1589:is a power of
1565:
1562:
1559:
1556:
1553:
1549:
1545:
1541:
1536:
1511:
1507:
1503:
1498:
1472:
1468:
1464:
1459:
1422:
1419:
1405:
1391:. Thus, every
1375:
1350:
1315:
1312:
1297:
1292:
1269:
1203:
1200:
1041:
1037:
1033:
1028:
1007:
1004:
1000:
966:
962:
958:
953:
924:
902:
876:
849:
846:
843:
840:
837:
834:
830:
826:
823:
820:
815:
810:
779:
774:
687:
683:
679:
674:
655:division rings
571:
568:
567:
566:
522:
507:exists unless
424:
407:
403:
399:
394:
361:natural number
357:
338:
324:of the unique
307:
286:natural number
274:
271:
241:
240:
229:
226:
219: summands
216:
210:
206:
203:
200:
197:
194:
170:additive group
168:of the ring's
157:
154:
140:
139:
128:
125:
118: summands
115:
109:
105:
102:
99:
96:
93:
25:characteristic
15:
9:
6:
4:
3:
2:
1843:
1832:
1829:
1827:
1824:
1823:
1821:
1809:
1803:
1800:. p. 4.
1799:
1795:
1790:
1789:
1776:
1772:
1768:
1764:
1760:
1754:
1746:
1742:
1741:
1733:
1731:
1726:
1712:
1708:
1701:
1694:
1687:
1683:
1657:
1651:
1647:
1639:
1637:
1633:
1630:it is also a
1612:
1608:
1593:
1587:
1582:
1577:
1557:
1543:
1539:
1526:
1505:
1501:
1487:
1466:
1462:
1448:
1443:
1440:
1433:
1428:
1418:
1394:
1365:
1340:
1336:
1335:ordered field
1331:
1329:
1328:p-adic fields
1325:
1321:
1311:
1295:
1258:
1254:
1246:
1241:
1235:
1225:
1221:
1217:
1213:
1199:
1197:
1193:
1188:
1183:
1182:
1176:
1172:
1168:
1163:
1159:
1153:
1148:
1143:
1137:
1131:
1125:
1121:
1117:
1113:
1106:
1102:
1097:
1091:
1087:
1081:
1075:
1069:
1063:
1057:
1035:
1031:
1005:
988:
982:
960:
956:
941:
891:
864:
841:
835:
828:
821:
813:
799:
798:quotient ring
777:
762:
756:
752:
748:is prime and
746:
740:
734:
728:
723:
718:
712:
706:
702:
681:
677:
662:
661:is infinite.
656:
653:, and to all
652:
648:
644:
636:
635:zero divisors
631:
622:
613:
607:
601:
597:
593:
589:
584:
578:
570:Case of rings
563:
557:
553:implies that
551:
547:
540:
534:
528:
523:
519:
512:
505:
501:
497:
491:
484:
478:
472:
468:
453:⋅ 1 = 0
452:
446:
433:
425:
422:
401:
397:
384:
380:
377:
372:
366:
362:
358:
354:
327:
323:
297:
291:
287:
282:
277:
276:
270:
268:
264:
263:
258:
253:
247:
227:
224:
214:
208:
204:
201:
198:
195:
192:
181:
180:
179:
176:
171:
167:
162:
153:
146:
126:
123:
113:
107:
103:
100:
97:
94:
91:
80:
79:
78:
75:
68:
61:
55:
47:
41:
34:
30:
26:
22:
1793:
1766:
1753:
1739:
1710:
1706:
1699:
1693:
1650:
1632:vector space
1591:
1585:
1578:
1444:
1438:
1431:
1427:finite field
1424:
1364:real numbers
1332:
1319:
1317:
1248:
1239:
1236:
1223:
1219:
1215:
1211:
1205:
1186:
1179:
1174:
1170:
1161:
1157:
1151:
1141:
1135:
1129:
1123:
1119:
1115:
1111:
1104:
1100:
1095:
1092:
1085:
1083:times gives
1079:
1073:
1067:
1061:
1055:
986:
980:
942:
862:
754:
750:
744:
738:
732:
726:
716:
710:
704:
699:of integers
663:
629:
611:
605:
599:
595:
582:
576:
573:
561:
555:
549:
545:
538:
532:
526:
517:
510:
503:
499:
495:
489:
482:
470:
466:
450:
444:
370:
364:
352:
295:
289:
280:
260:
251:
245:
242:
174:
163:
159:
148:exists, and
144:
141:
77:such that:
73:
66:
62:
39:
32:
24:
18:
1826:Ring theory
1581:finite ring
1251:prime field
383:factor ring
374:contains a
178:such that:
152:otherwise.
21:mathematics
1820:Categories
1722:References
1237:Any field
1155:. The map
1077:to itself
448:for which
379:isomorphic
368:such that
293:such that
156:Motivation
1196:injective
1003:→
892:contains
664:The ring
649:, to all
621:zero ring
209:⏟
199:⋯
108:⏟
98:⋯
63:That is,
1761:(2003).
1333:For any
1245:subfield
1059:divides
543:for all
514:divides
498: :
166:exponent
1785:Sources
1684:of the
730:, then
722:subring
619:is the
475:is the
381:to the
376:subring
320:is the
284:is the
1804:
1680:is an
1484:, the
1326:. The
1259:field
1222:. The
1194:it is
1190:is an
759:is an
701:modulo
647:fields
322:kernel
23:, the
1642:Notes
1449:over
1184:. If
792:with
720:is a
714:. If
643:prime
588:rings
516:char
509:char
488:char
481:char
465:char(
421:image
328:from
259:(see
65:char(
38:char(
27:of a
1802:ISBN
1709:) =
1425:The
1133:and
1118:) =
1099:has
736:and
586:are
580:and
486:and
430:are
257:rngs
29:ring
1771:doi
1488:of
1430:GF(
1218:or
1214:or
1139:in
1088:= 0
936:is
888:of
724:of
641:or
574:If
541:= 0
530:is
479:of
350:to
19:In
1822::
1796:.
1765:.
1729:^
1658:,
1576:.
1442:.
1198:.
1173:→
1160:↦
1122:+
1114:+
1090:.
1086:nr
943:A
940:.
598:→
548:∈
539:ka
502:→
469:×
269:.
1810:.
1777:.
1773::
1747:.
1714:.
1711:p
1707:p
1705:(
1700:p
1667:Z
1617:Z
1613:p
1609:/
1604:Z
1592:p
1586:p
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1561:)
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1555:(
1552:(
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1540:/
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1502:/
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1471:Z
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1434:)
1432:p
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1389:0
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1291:F
1268:Q
1240:F
1232:0
1228:1
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1175:R
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1162:x
1158:x
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1142:R
1136:y
1130:x
1124:y
1120:x
1116:y
1112:x
1110:(
1105:p
1096:R
1080:n
1074:r
1068:r
1062:n
1056:R
1040:Z
1036:n
1032:/
1027:Z
1006:R
999:Z
987:R
981:n
965:Z
961:n
957:/
952:Z
938:0
923:C
901:Z
875:C
863:p
848:)
845:)
842:X
839:(
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833:(
829:/
825:]
822:X
819:[
814:p
809:F
794:p
778:p
773:F
757:)
755:X
753:(
751:q
745:p
739:S
733:R
727:S
717:R
711:n
705:n
686:Z
682:n
678:/
673:Z
659:0
639:0
630:R
625:0
617:1
612:R
606:S
600:S
596:R
583:S
577:R
565:.
562:n
556:k
550:R
546:a
533:n
527:R
521:.
518:A
511:B
504:B
500:A
496:f
490:B
483:A
473:)
471:B
467:A
461:0
457:0
451:n
445:n
440:0
436:1
406:Z
402:n
398:/
393:Z
371:R
365:n
356:.
353:R
337:Z
306:Z
296:n
290:n
281:R
252:n
246:a
228:0
225:=
215:n
205:a
202:+
196:+
193:a
175:n
150:0
145:n
127:0
124:=
114:n
104:1
101:+
95:+
92:1
74:n
69:)
67:R
58:0
56:(
50:1
48:(
42:)
40:R
33:R
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