249:
148:
171:
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.
1585:
1321:
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.
1639:
1532:
1493:
1062:
987:
708:
428:
1749:
869:
1649:
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.
1027:
1319:
801:
1689:
1426:
1396:
1371:
1290:
945:
923:
897:
359:
328:
1665:
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
1341:
are characteristic zero fields that are widely used in number theory. They have absolute values which are very different from those of complex numbers.
272:
195:
94:
1816:
626:. This can sometimes be used to exclude the possibility of certain ring homomorphisms. The only ring with characteristic
1540:
1699:. Again this applies when a ring has a multiplicative identity element (which is preserved by ring homomorphisms).
1609:
1502:
1463:
1032:
957:
678:
398:
813:
1841:
17:
1004:
1295:
777:
1672:
1409:
1379:
1354:
1273:
928:
906:
880:
342:
311:
176:
1773:
1403:
1191:
771:
487:
442:
1836:
332:
8:
1535:
1157:
657:
431:
1808:
711:
598:
39:
1812:
1755:
1696:
1496:
1457:
1177:
602:
389:
336:
64:
1781:
1769:
1456:
There exist infinite fields of prime characteristic. For example, the field of all
277:
56:
1666:
1349:
1267:
1255:
1202:
661:
1692:
1646:
1334:
900:
371:
296:
180:
1785:
266:
exists; otherwise zero). This definition applies in the more general class of
1830:
1345:
808:
665:
267:
1708:
It is a vector space over a finite field, which we have shown to be of size
453:
is the largest. Then the characteristic of a ring is the smallest value of
71:). If no such number exists, the ring is said to have characteristic zero.
1642:
1437:
1338:
1221:
or a prime number. A field of non-zero characteristic is called a field of
653:
645:
1591:
1374:
474:. This is the appropriate partial ordering because of such facts as that
393:
31:
55:, is defined to be the smallest positive number of copies of the ring's
276:); for (unital) rings the two definitions are equivalent due to their
1206:
631:
273:
Ring (mathematics) § Multiplicative identity and the term "ring"
244:{\displaystyle \underbrace {a+\cdots +a} _{n{\text{ summands}}}=0}
143:{\displaystyle \underbrace {1+\cdots +1} _{n{\text{ summands}}}=0}
732:
386:
1217:
As mentioned above, the characteristic of any field is either
989:-algebra is equivalently a ring whose characteristic divides
1245:; otherwise it has the same value as the characteristic.
1774:"5. Characteristic exponent of a field. Perfect fields"
1743:
1741:
1675:
1612:
1543:
1505:
1466:
1412:
1382:
1357:
1298:
1276:
1035:
1007:
960:
931:
909:
883:
816:
780:
681:
401:
345:
314:
198:
97:
1738:
1683:
1633:
1579:
1526:
1487:
1420:
1390:
1365:
1313:
1284:
1056:
1021:
981:
939:
917:
891:
863:
795:
702:
422:
353:
322:
243:
142:
1431:
1237:is defined similarly, except that it is equal to
27:Smallest integer n for which n equals 0 in a ring
1828:
1324:
1580:{\displaystyle \mathbb {Z} /p\mathbb {Z} ((T))}
466:. If nothing "smaller" (in this ordering) than
283:
175:The characteristic may also be taken to be the
753:have the same characteristic. For example, if
470:will suffice, then the characteristic is
1747:
1266:. This subfield is isomorphic to either the
1748:Fraleigh, John B.; Brand, Neal E. (2020).
1634:{\displaystyle \mathbb {Z} /p\mathbb {Z} }
1527:{\displaystyle \mathbb {Z} /p\mathbb {Z} }
1488:{\displaystyle \mathbb {Z} /p\mathbb {Z} }
1057:{\displaystyle \mathbb {Z} /n\mathbb {Z} }
982:{\displaystyle \mathbb {Z} /n\mathbb {Z} }
703:{\displaystyle \mathbb {Z} /n\mathbb {Z} }
423:{\displaystyle \mathbb {Z} /n\mathbb {Z} }
1677:
1627:
1614:
1558:
1545:
1520:
1507:
1481:
1468:
1414:
1384:
1359:
1301:
1278:
1050:
1037:
1009:
975:
962:
933:
911:
885:
819:
783:
696:
683:
416:
403:
347:
316:
183:, that is, the smallest positive integer
1768:
864:{\displaystyle \mathbb {F} _{p}/(q(X))}
14:
1829:
1802:
1064:if and only if the characteristic of
656:. In particular, this applies to all
648:, then its characteristic is either
24:
1751:A First Course in Abstract Algebra
634:, which has only a single element
25:
1853:
1606:. Since in that case it contains
1406:and the field of complex numbers
1212:
1022:{\displaystyle \mathbb {Z} \to R}
995:. This is because for every ring
1314:{\displaystyle \mathbb {F} _{p}}
796:{\displaystyle \mathbb {F} _{p}}
580:
504:, and that no ring homomorphism
82:is the smallest positive number
1029:, and this map factors through
774:with coefficients in the field
437:When the non-negative integers
1762:
1702:
1659:
1574:
1571:
1565:
1562:
1432:Fields of prime characteristic
1013:
858:
855:
849:
843:
835:
829:
620:divides the characteristic of
13:
1:
1732:
1325:Fields of characteristic zero
1001:there is a ring homomorphism
877:. Another example: The field
871:is a field of characteristic
668:. Any ring of characteristic
644:does not have any nontrivial
614:, then the characteristic of
535:The characteristic of a ring
289:The characteristic of a ring
166:
1684:{\displaystyle \mathbb {Z} }
1428:are of characteristic zero.
1421:{\displaystyle \mathbb {C} }
1391:{\displaystyle \mathbb {R} }
1366:{\displaystyle \mathbb {Q} }
1285:{\displaystyle \mathbb {Q} }
940:{\displaystyle \mathbb {C} }
918:{\displaystyle \mathbb {Z} }
892:{\displaystyle \mathbb {C} }
354:{\displaystyle \mathbb {Z} }
323:{\displaystyle \mathbb {Z} }
284:Equivalent characterizations
7:
1780:. Springer. p. A.V.7.
1241:when the characteristic is
925:, so the characteristic of
547:precisely if the statement
10:
1858:
1795:
1645:over that field, and from
1329:The most common fields of
1156:– the normally incorrect "
434:of the above homomorphism.
370:The characteristic is the
1786:10.1007/978-3-642-61698-3
1333:are the subfields of the
1082:in the ring, then adding
1803:McCoy, Neal H. (1973) .
1778:Algebra II, Chapters 4–7
1652:
1594:of prime characteristic
1398:, the characteristic is
1348:, such as the field of
1235:characteristic exponent
1227:positive characteristic
1076:. In this case for any
638:. If a nontrivial ring
260:of the ring (again, if
63:) that will sum to the
57:multiplicative identity
1685:
1635:
1581:
1528:
1489:
1422:
1404:algebraic number field
1392:
1367:
1315:
1286:
1192:Frobenius homomorphism
1189:, which is called the
1104:If a commutative ring
1058:
1023:
983:
941:
919:
893:
865:
797:
772:irreducible polynomial
704:
445:by divisibility, then
424:
355:
324:
245:
144:
1686:
1636:
1582:
1536:formal Laurent series
1529:
1490:
1423:
1393:
1368:
1316:
1287:
1254:has a unique minimal
1223:finite characteristic
1059:
1024:
984:
942:
920:
894:
866:
798:
705:
488:least common multiple
425:
356:
325:
246:
145:
1673:
1610:
1541:
1503:
1464:
1410:
1380:
1355:
1296:
1274:
1231:prime characteristic
1112:prime characteristic
1033:
1005:
958:
929:
907:
881:
814:
778:
679:
449:is the smallest and
399:
343:
312:
196:
95:
1842:Field (mathematics)
1805:The Theory of Rings
1447:has characteristic
1331:characteristic zero
807:elements, then the
719:has characteristic
601:and there exists a
1809:Chelsea Publishing
1681:
1631:
1577:
1524:
1485:
1458:rational functions
1418:
1388:
1363:
1311:
1292:or a finite field
1282:
1258:, also called its
1160:" holds for power
1054:
1019:
979:
937:
915:
889:
861:
793:
700:
420:
351:
320:
254:for every element
241:
234:
222:
140:
133:
121:
1818:978-0-8284-0266-8
1770:Bourbaki, Nicolas
1756:Pearson Education
1714:, so its size is
1697:category of rings
1497:algebraic closure
1178:ring homomorphism
1138:for all elements
603:ring homomorphism
570:is a multiple of
443:partially ordered
439:{0, 1, 2, 3, ...}
337:ring homomorphism
231:
201:
199:
153:if such a number
130:
100:
98:
65:additive identity
16:(Redirected from
1849:
1822:
1790:
1789:
1766:
1760:
1759:
1754:(8th ed.).
1745:
1726:
1724:
1713:
1706:
1700:
1690:
1688:
1687:
1682:
1680:
1663:
1640:
1638:
1637:
1632:
1630:
1622:
1617:
1605:
1599:
1590:The size of any
1586:
1584:
1583:
1578:
1561:
1553:
1548:
1534:or the field of
1533:
1531:
1530:
1525:
1523:
1515:
1510:
1494:
1492:
1491:
1486:
1484:
1476:
1471:
1452:
1446:
1427:
1425:
1424:
1419:
1417:
1401:
1397:
1395:
1394:
1389:
1387:
1373:or the field of
1372:
1370:
1369:
1364:
1362:
1350:rational numbers
1320:
1318:
1317:
1312:
1310:
1309:
1304:
1291:
1289:
1288:
1283:
1281:
1264:
1263:
1253:
1244:
1240:
1220:
1200:
1188:
1175:
1165:
1158:freshman's dream
1155:
1149:
1143:
1137:
1118:
1109:
1100:
1093:
1087:
1081:
1075:
1069:
1063:
1061:
1060:
1055:
1053:
1045:
1040:
1028:
1026:
1025:
1020:
1012:
1000:
994:
988:
986:
985:
980:
978:
970:
965:
950:
946:
944:
943:
938:
936:
924:
922:
921:
916:
914:
898:
896:
895:
890:
888:
876:
870:
868:
867:
862:
842:
828:
827:
822:
806:
802:
800:
799:
794:
792:
791:
786:
769:
758:
752:
746:
740:
730:
724:
718:
709:
707:
706:
701:
699:
691:
686:
671:
662:integral domains
651:
643:
637:
629:
625:
619:
613:
596:
590:
575:
569:
563:
553:
546:
540:
531:
524:
517:
503:
496:
485:
473:
469:
465:
458:
452:
448:
440:
429:
427:
426:
421:
419:
411:
406:
384:
378:
366:
360:
358:
357:
352:
350:
330:
329:
327:
326:
321:
319:
303:
294:
278:distributive law
265:
259:
250:
248:
247:
242:
233:
232:
229:
223:
218:
188:
162:
158:
149:
147:
146:
141:
132:
131:
128:
122:
117:
87:
81:
70:
62:
54:
47:, often denoted
46:
21:
1857:
1856:
1852:
1851:
1850:
1848:
1847:
1846:
1827:
1826:
1825:
1819:
1798:
1793:
1767:
1763:
1746:
1739:
1735:
1730:
1729:
1715:
1709:
1707:
1703:
1676:
1674:
1671:
1670:
1667:category theory
1664:
1660:
1655:
1626:
1618:
1613:
1611:
1608:
1607:
1601:
1595:
1557:
1549:
1544:
1542:
1539:
1538:
1519:
1511:
1506:
1504:
1501:
1500:
1480:
1472:
1467:
1465:
1462:
1461:
1448:
1440:
1434:
1413:
1411:
1408:
1407:
1399:
1383:
1381:
1378:
1377:
1358:
1356:
1353:
1352:
1335:complex numbers
1327:
1305:
1300:
1299:
1297:
1294:
1293:
1277:
1275:
1272:
1271:
1268:rational number
1261:
1260:
1249:
1242:
1238:
1218:
1215:
1213:Case of fields
1203:integral domain
1196:
1180:
1176:then defines a
1167:
1161:
1151:
1145:
1139:
1120:
1119:, then we have
1114:
1105:
1095:
1089:
1083:
1077:
1071:
1065:
1049:
1041:
1036:
1034:
1031:
1030:
1008:
1006:
1003:
1002:
996:
990:
974:
966:
961:
959:
956:
955:
948:
932:
930:
927:
926:
910:
908:
905:
904:
901:complex numbers
884:
882:
879:
878:
872:
838:
823:
818:
817:
815:
812:
811:
804:
787:
782:
781:
779:
776:
775:
760:
754:
748:
742:
736:
726:
720:
714:
695:
687:
682:
680:
677:
676:
669:
649:
639:
635:
627:
621:
615:
605:
592:
586:
583:
571:
565:
555:
548:
542:
536:
526:
519:
505:
498:
491:
475:
471:
467:
460:
454:
450:
446:
438:
430:, which is the
415:
407:
402:
400:
397:
396:
380:
374:
362:
346:
344:
341:
340:
315:
313:
310:
309:
305:
299:
290:
286:
261:
255:
228:
224:
202:
200:
197:
194:
193:
184:
169:
160:
154:
127:
123:
101:
99:
96:
93:
92:
83:
75:
68:
60:
48:
42:
28:
23:
22:
15:
12:
11:
5:
1855:
1845:
1844:
1839:
1824:
1823:
1817:
1799:
1797:
1794:
1792:
1791:
1761:
1736:
1734:
1731:
1728:
1727:
1701:
1693:initial object
1679:
1657:
1656:
1654:
1651:
1647:linear algebra
1629:
1625:
1621:
1616:
1600:is a power of
1576:
1573:
1570:
1567:
1564:
1560:
1556:
1552:
1547:
1522:
1518:
1514:
1509:
1483:
1479:
1475:
1470:
1433:
1430:
1416:
1402:. Thus, every
1386:
1361:
1326:
1323:
1308:
1303:
1280:
1214:
1211:
1052:
1048:
1044:
1039:
1018:
1015:
1011:
977:
973:
969:
964:
935:
913:
887:
860:
857:
854:
851:
848:
845:
841:
837:
834:
831:
826:
821:
790:
785:
698:
694:
690:
685:
666:division rings
582:
579:
578:
577:
533:
518:exists unless
435:
418:
414:
410:
405:
372:natural number
368:
349:
335:of the unique
318:
297:natural number
285:
282:
252:
251:
240:
237:
230: summands
227:
221:
217:
214:
211:
208:
205:
181:additive group
179:of the ring's
168:
165:
151:
150:
139:
136:
129: summands
126:
120:
116:
113:
110:
107:
104:
36:characteristic
26:
9:
6:
4:
3:
2:
1854:
1843:
1840:
1838:
1835:
1834:
1832:
1820:
1814:
1811:. p. 4.
1810:
1806:
1801:
1800:
1787:
1783:
1779:
1775:
1771:
1765:
1757:
1753:
1752:
1744:
1742:
1737:
1723:
1719:
1712:
1705:
1698:
1694:
1668:
1662:
1658:
1650:
1648:
1644:
1641:it is also a
1623:
1619:
1604:
1598:
1593:
1588:
1568:
1554:
1550:
1537:
1516:
1512:
1498:
1477:
1473:
1459:
1454:
1451:
1444:
1439:
1429:
1405:
1376:
1351:
1347:
1346:ordered field
1342:
1340:
1339:p-adic fields
1336:
1332:
1322:
1306:
1269:
1265:
1257:
1252:
1246:
1236:
1232:
1228:
1224:
1210:
1208:
1204:
1199:
1194:
1193:
1187:
1183:
1179:
1174:
1170:
1164:
1159:
1154:
1148:
1142:
1136:
1132:
1128:
1124:
1117:
1113:
1108:
1102:
1098:
1092:
1086:
1080:
1074:
1068:
1046:
1042:
1016:
999:
993:
971:
967:
952:
902:
875:
852:
846:
839:
832:
824:
810:
809:quotient ring
788:
773:
767:
763:
759:is prime and
757:
751:
745:
739:
734:
729:
723:
717:
713:
692:
688:
673:
672:is infinite.
667:
664:, and to all
663:
659:
655:
647:
646:zero divisors
642:
633:
624:
618:
612:
608:
604:
600:
595:
589:
581:Case of rings
574:
568:
564:implies that
562:
558:
551:
545:
539:
534:
530:
523:
516:
512:
508:
502:
495:
489:
483:
479:
464:⋅ 1 = 0
463:
457:
444:
436:
433:
412:
408:
395:
391:
388:
383:
377:
373:
369:
365:
338:
334:
308:
302:
298:
293:
288:
287:
281:
279:
275:
274:
269:
264:
258:
238:
235:
225:
219:
215:
212:
209:
206:
203:
192:
191:
190:
187:
182:
178:
173:
164:
157:
137:
134:
124:
118:
114:
111:
108:
105:
102:
91:
90:
89:
86:
79:
72:
66:
58:
52:
45:
41:
37:
33:
19:
1804:
1777:
1764:
1750:
1721:
1717:
1710:
1704:
1661:
1643:vector space
1602:
1596:
1589:
1455:
1449:
1442:
1438:finite field
1435:
1375:real numbers
1343:
1330:
1328:
1259:
1250:
1247:
1234:
1230:
1226:
1222:
1216:
1197:
1190:
1185:
1181:
1172:
1168:
1162:
1152:
1146:
1140:
1134:
1130:
1126:
1122:
1115:
1111:
1106:
1103:
1096:
1094:times gives
1090:
1084:
1078:
1072:
1066:
997:
991:
953:
873:
765:
761:
755:
749:
743:
737:
727:
721:
715:
710:of integers
674:
640:
622:
616:
610:
606:
593:
587:
584:
572:
566:
560:
556:
549:
543:
537:
528:
521:
514:
510:
506:
500:
493:
481:
477:
461:
455:
381:
375:
363:
306:
300:
291:
271:
262:
256:
253:
185:
174:
170:
159:exists, and
155:
152:
88:such that:
84:
77:
73:
50:
43:
35:
29:
1837:Ring theory
1592:finite ring
1262:prime field
394:factor ring
385:contains a
189:such that:
163:otherwise.
32:mathematics
18:Prime field
1831:Categories
1733:References
1248:Any field
1166:. The map
1088:to itself
459:for which
390:isomorphic
379:such that
304:such that
167:Motivation
1207:injective
1014:→
903:contains
675:The ring
660:, to all
632:zero ring
220:⏟
210:⋯
119:⏟
109:⋯
74:That is,
1772:(2003).
1344:For any
1256:subfield
1070:divides
554:for all
525:divides
509: :
177:exponent
1796:Sources
1695:of the
741:, then
733:subring
630:is the
486:is the
392:to the
387:subring
331:is the
295:is the
1815:
1691:is an
1495:, the
1337:. The
1270:field
1233:. The
1205:it is
1201:is an
770:is an
712:modulo
658:fields
333:kernel
34:, the
1653:Notes
1460:over
1195:. If
803:with
731:is a
725:. If
654:prime
599:rings
527:char
520:char
499:char
492:char
476:char(
432:image
339:from
270:(see
76:char(
49:char(
38:of a
1813:ISBN
1720:) =
1436:The
1144:and
1129:) =
1110:has
747:and
597:are
591:and
497:and
441:are
268:rngs
40:ring
1782:doi
1499:of
1441:GF(
1229:or
1225:or
1150:in
1099:= 0
947:is
899:of
735:of
652:or
585:If
552:= 0
541:is
490:of
361:to
30:In
1833::
1807:.
1776:.
1740:^
1669:,
1587:.
1453:.
1209:.
1184:→
1171:↦
1133:+
1125:+
1101:.
1097:nr
954:A
951:.
609:→
559:∈
550:ka
513:→
480:×
280:.
1821:.
1788:.
1784::
1758:.
1725:.
1722:p
1718:p
1716:(
1711:p
1678:Z
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1121:(
1116:p
1107:R
1091:n
1085:r
1079:r
1073:n
1067:R
1051:Z
1047:n
1043:/
1038:Z
1017:R
1010:Z
998:R
992:n
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912:Z
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859:)
856:)
853:X
850:(
847:q
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840:/
836:]
833:X
830:[
825:p
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768:)
766:X
764:(
762:q
756:p
750:S
744:R
738:S
728:R
722:n
716:n
697:Z
693:n
689:/
684:Z
670:0
650:0
641:R
636:0
628:1
623:R
617:S
611:S
607:R
594:S
588:R
576:.
573:n
567:k
561:R
557:a
544:n
538:R
532:.
529:A
522:B
515:B
511:A
507:f
501:B
494:A
484:)
482:B
478:A
472:0
468:0
462:n
456:n
451:0
447:1
417:Z
413:n
409:/
404:Z
382:R
376:n
367:.
364:R
348:Z
317:Z
307:n
301:n
292:R
263:n
257:a
239:0
236:=
226:n
216:a
213:+
207:+
204:a
186:n
161:0
156:n
138:0
135:=
125:n
115:1
112:+
106:+
103:1
85:n
80:)
78:R
69:0
67:(
61:1
59:(
53:)
51:R
44:R
20:)
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