25:
1179:
336:
1036:
829:
494:
1484:
1812:
573:
623:
923:
866:
1371:
364:
684:
656:
524:
1600:
1575:
1550:
1528:
1506:
1447:
1425:
1403:
1337:
1300:
1278:
1248:
1226:
1087:
1057:
963:
888:
753:
731:
706:
412:
390:
241:
371:
of the polynomial. Two such polynomials are equal only if the corresponding coefficients are equal. In contrast, two polynomial functions in a variable
756:
by substitution. But the distinction is important because information may be lost when this substitution is made. For example, when working in
133:
that is used formally, without reference to any value. In other words, this is just a symbol used in a formal way. Indeterminates occur in
89:
42:
1060:. This is similar to the definition of a polynomial, except that an infinite number of the coefficients may be nonzero. Unlike the
61:
1094:
246:
179:, while indeterminates do not represent anything. Many authors do not distinguish indeterminates from other sorts of variables.
68:
152:
A fundamental property of an indeterminate is that it can be substituted with any mathematical expressions to which the same
1740:
75:
1833:
1709:
1680:
1754:
1723:
969:
108:
57:
766:
1822:
424:
46:
1655:
1873:
1454:
163:
is relatively recent, and was initially introduced for distinguishing a polynomial from its associated
1068:
are irrelevant (since there is no function at play). So power series that would diverge for values of
1863:
1065:
533:
171:. The main difference is that a free variable is intended to represent a unspecified element of some
82:
142:
1191:
926:
is not the zero polynomial, since the coefficients, 0, 1 and −1, respectively, are not all zero.
153:
35:
1612:
1201:
584:
198:
130:
1256:
895:
838:
206:
1868:
1622:
1344:
342:
172:
146:
8:
1205:
941:
935:
663:
635:
503:
164:
138:
1582:
1557:
1715:
1617:
1535:
1513:
1491:
1432:
1410:
1388:
1307:
1285:
1263:
1233:
1211:
1072:
1042:
948:
873:
757:
738:
716:
691:
397:
375:
226:
191:
1783:
1843:
1818:
1750:
1719:
1690:
1661:
1603:
are not necessarily identical (since free algebra is by definition non-commutative).
1382:
1197:
183:
1839:
1686:
1252:
168:
1857:
1746:
1378:
1061:
368:
176:
122:
578:
are unequal, since 2 does not equal 5, and 3 does not equal 2. In fact,
1665:
218:
209:
and every nonconstant polynomial must be considered as indeterminates.
134:
24:
891:
having any value in the modulo-2 system. However, the polynomial
1847:
1694:
1174:{\displaystyle 1+x+2x^{2}+6x^{3}+\ldots +n!x^{n}+\ldots \,}
331:{\displaystyle a_{0}+a_{1}X+a_{2}X^{2}+\ldots +a_{n}X^{n}}
1340:
also uses these operations, and convention holds that
1585:
1560:
1538:
1516:
1494:
1457:
1435:
1413:
1391:
1347:
1310:
1288:
1266:
1236:
1214:
1097:
1075:
1045:
972:
951:
898:
876:
841:
769:
741:
719:
694:
666:
638:
587:
536:
506:
427:
400:
378:
345:
249:
229:
156:
apply as the operations aplied to the indeterminate.
1653:
1259:
as operations. In particular, if two indeterminates
49:. Unsourced material may be challenged and removed.
1594:
1569:
1544:
1522:
1500:
1478:
1441:
1419:
1397:
1365:
1331:
1294:
1272:
1242:
1220:
1173:
1081:
1051:
1030:
957:
917:
882:
860:
823:
747:
725:
700:
678:
650:
617:
567:
518:
488:
406:
384:
358:
330:
235:
712:The distinction is subtle, since a polynomial in
527:and not equal otherwise. But the two polynomials
1855:
1031:{\displaystyle a_{0}+a_{1}X+a_{2}X^{2}+\ldots }
1377:Indeterminates may also be used to generate a
1229:, the set of polynomials with coefficients in
205:. This uncommon definition implies that every
393:may be equal or not at a particular value of
1473:
1461:
1038:, where no value is assigned to the symbol
824:{\displaystyle 0-0^{2}=0,\quad 1-1^{2}=0,}
709:is not, and does not designate, a number.
1170:
489:{\displaystyle f(x)=2+3x,\quad g(x)=5+2x}
109:Learn how and when to remove this message
16:Symbol treated as a mathematical variable
1810:
1769:
1406:. For instance, with two indeterminates
1711:An Introduction to Algebraic Structures
197:as an element of a larger ring that is
1856:
1738:
1707:
1257:polynomial addition and multiplication
1064:encountered in calculus, questions of
929:
1831:
1781:
1678:
1641:
1479:{\displaystyle A\langle X,Y\rangle }
47:adding citations to reliable sources
18:
1303:are used, then the polynomial ring
13:
1553:, and with the understanding that
14:
1885:
223:A polynomial in an indeterminate
1185:
734:can be changed to a function in
23:
795:
568:{\displaystyle 2+3X,\quad 5+2X}
552:
458:
145:that are viewed as independent
34:needs additional citations for
1835:Introduction To Modern Algebra
1775:
1763:
1742:Introduction to Modern Algebra
1732:
1701:
1672:
1647:
1635:
1326:
1314:
869:is identically equal to 0 for
468:
462:
437:
431:
212:
1:
1804:
1654:Joseph Miller Thomas (1974).
1196:Indeterminates are useful in
966:is an expression of the form
243:is an expression of the form
58:"Indeterminate" variable
1487:includes sums of strings in
7:
1606:
834:so the polynomial function
418:For example, the functions
10:
1890:
1189:
933:
216:
167:. Indeterminates resemble
141:, and, more generally, in
1679:Lewis, Donald J. (1965).
618:{\displaystyle 2+3X=a+bX}
1811:Herstein, I. N. (1975).
1628:
1832:McCoy, Neal H. (1960),
1739:Marcus, Marvin (1978).
1708:Landin, Joseph (1989).
1682:Introduction to Algebra
1531:, with coefficients in
1204:. For example, given a
1202:mathematical structures
1192:Generator (mathematics)
918:{\displaystyle X-X^{2}}
861:{\displaystyle x-x^{2}}
1660:. William Byrd Press.
1613:Indeterminate equation
1596:
1571:
1546:
1524:
1502:
1480:
1443:
1421:
1399:
1367:
1333:
1296:
1274:
1244:
1222:
1175:
1083:
1053:
1032:
959:
919:
884:
862:
825:
749:
727:
702:
680:
652:
619:
569:
520:
490:
408:
386:
360:
332:
237:
1788:mathworld.wolfram.com
1784:"Formal Power Series"
1597:
1572:
1547:
1525:
1503:
1481:
1444:
1422:
1400:
1368:
1366:{\displaystyle XY=YX}
1334:
1297:
1275:
1245:
1223:
1176:
1084:
1054:
1033:
960:
920:
885:
863:
826:
750:
728:
703:
681:
653:
620:
570:
521:
491:
409:
387:
361:
359:{\displaystyle a_{i}}
333:
238:
207:transcendental number
1644:, pp. 189, 190)
1623:Indeterminate system
1583:
1558:
1536:
1514:
1492:
1455:
1433:
1411:
1389:
1345:
1308:
1286:
1264:
1234:
1212:
1095:
1073:
1043:
970:
949:
944:in an indeterminate
896:
874:
839:
767:
739:
717:
692:
664:
636:
585:
534:
504:
425:
398:
376:
343:
247:
227:
186:textbooks define an
147:mathematical objects
43:improve this article
1874:Mathematical series
1782:Weisstein, Eric W.
1749:. p. 140–141.
1450:, the free algebra
942:formal power series
936:Formal power series
930:Formal power series
679:{\displaystyle b=3}
651:{\displaystyle a=2}
519:{\displaystyle x=3}
165:polynomial function
139:formal power series
1716:Dover Publications
1618:Indeterminate form
1595:{\displaystyle YX}
1592:
1570:{\displaystyle XY}
1567:
1542:
1520:
1498:
1476:
1439:
1417:
1395:
1363:
1329:
1292:
1270:
1240:
1218:
1171:
1079:
1049:
1028:
955:
915:
880:
858:
821:
745:
723:
698:
687:. This is because
676:
648:
615:
565:
516:
486:
404:
382:
356:
328:
233:
159:The concept of an
1814:Topics in Algebra
1657:A Primer On Roots
1545:{\displaystyle A}
1523:{\displaystyle Y}
1501:{\displaystyle X}
1442:{\displaystyle Y}
1420:{\displaystyle X}
1398:{\displaystyle A}
1332:{\displaystyle K}
1295:{\displaystyle Y}
1273:{\displaystyle X}
1243:{\displaystyle K}
1221:{\displaystyle K}
1082:{\displaystyle x}
1052:{\displaystyle X}
958:{\displaystyle X}
883:{\displaystyle x}
748:{\displaystyle x}
726:{\displaystyle X}
701:{\displaystyle X}
407:{\displaystyle x}
385:{\displaystyle x}
236:{\displaystyle X}
119:
118:
111:
93:
1881:
1864:Abstract algebra
1850:
1828:
1798:
1797:
1795:
1794:
1779:
1773:
1767:
1761:
1760:
1736:
1730:
1729:
1705:
1699:
1698:
1687:Harper & Row
1676:
1670:
1669:
1651:
1645:
1639:
1601:
1599:
1598:
1593:
1576:
1574:
1573:
1568:
1551:
1549:
1548:
1543:
1529:
1527:
1526:
1521:
1507:
1505:
1504:
1499:
1485:
1483:
1482:
1477:
1448:
1446:
1445:
1440:
1426:
1424:
1423:
1418:
1404:
1402:
1401:
1396:
1383:commutative ring
1372:
1370:
1369:
1364:
1338:
1336:
1335:
1330:
1301:
1299:
1298:
1293:
1279:
1277:
1276:
1271:
1249:
1247:
1246:
1241:
1227:
1225:
1224:
1219:
1198:abstract algebra
1180:
1178:
1177:
1172:
1163:
1162:
1138:
1137:
1122:
1121:
1088:
1086:
1085:
1080:
1058:
1056:
1055:
1050:
1037:
1035:
1034:
1029:
1021:
1020:
1011:
1010:
995:
994:
982:
981:
964:
962:
961:
956:
924:
922:
921:
916:
914:
913:
889:
887:
886:
881:
867:
865:
864:
859:
857:
856:
830:
828:
827:
822:
811:
810:
785:
784:
760:, we have that:
754:
752:
751:
746:
732:
730:
729:
724:
707:
705:
704:
699:
685:
683:
682:
677:
657:
655:
654:
649:
624:
622:
621:
616:
574:
572:
571:
566:
525:
523:
522:
517:
495:
493:
492:
487:
413:
411:
410:
405:
391:
389:
388:
383:
365:
363:
362:
357:
355:
354:
337:
335:
334:
329:
327:
326:
317:
316:
298:
297:
288:
287:
272:
271:
259:
258:
242:
240:
239:
234:
204:
196:
184:abstract algebra
182:Some authors of
114:
107:
103:
100:
94:
92:
51:
27:
19:
1889:
1888:
1884:
1883:
1882:
1880:
1879:
1878:
1854:
1853:
1840:Allyn and Bacon
1825:
1807:
1802:
1801:
1792:
1790:
1780:
1776:
1768:
1764:
1757:
1737:
1733:
1726:
1718:. p. 204.
1706:
1702:
1689:. p. 160.
1677:
1673:
1652:
1648:
1640:
1636:
1631:
1609:
1584:
1581:
1580:
1559:
1556:
1555:
1537:
1534:
1533:
1515:
1512:
1511:
1493:
1490:
1489:
1456:
1453:
1452:
1434:
1431:
1430:
1412:
1409:
1408:
1390:
1387:
1386:
1346:
1343:
1342:
1309:
1306:
1305:
1287:
1284:
1283:
1265:
1262:
1261:
1253:polynomial ring
1235:
1232:
1231:
1213:
1210:
1209:
1200:for generating
1194:
1188:
1182:, are allowed.
1158:
1154:
1133:
1129:
1117:
1113:
1096:
1093:
1092:
1074:
1071:
1070:
1044:
1041:
1040:
1016:
1012:
1006:
1002:
990:
986:
977:
973:
971:
968:
967:
950:
947:
946:
938:
932:
909:
905:
897:
894:
893:
875:
872:
871:
852:
848:
840:
837:
836:
806:
802:
780:
776:
768:
765:
764:
740:
737:
736:
718:
715:
714:
693:
690:
689:
665:
662:
661:
637:
634:
633:
586:
583:
582:
535:
532:
531:
505:
502:
501:
499:are equal when
426:
423:
422:
399:
396:
395:
377:
374:
373:
367:are called the
350:
346:
344:
341:
340:
322:
318:
312:
308:
293:
289:
283:
279:
267:
263:
254:
250:
248:
245:
244:
228:
225:
224:
221:
215:
202:
194:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
1887:
1877:
1876:
1871:
1866:
1852:
1851:
1829:
1823:
1806:
1803:
1800:
1799:
1774:
1772:, Section 3.9.
1762:
1755:
1731:
1724:
1700:
1671:
1646:
1633:
1632:
1630:
1627:
1626:
1625:
1620:
1615:
1608:
1605:
1591:
1588:
1566:
1563:
1541:
1519:
1497:
1475:
1472:
1469:
1466:
1463:
1460:
1438:
1416:
1394:
1362:
1359:
1356:
1353:
1350:
1328:
1325:
1322:
1319:
1316:
1313:
1291:
1269:
1239:
1217:
1190:Main article:
1187:
1184:
1169:
1166:
1161:
1157:
1153:
1150:
1147:
1144:
1141:
1136:
1132:
1128:
1125:
1120:
1116:
1112:
1109:
1106:
1103:
1100:
1078:
1048:
1027:
1024:
1019:
1015:
1009:
1005:
1001:
998:
993:
989:
985:
980:
976:
954:
934:Main article:
931:
928:
912:
908:
904:
901:
879:
855:
851:
847:
844:
832:
831:
820:
817:
814:
809:
805:
801:
798:
794:
791:
788:
783:
779:
775:
772:
744:
722:
697:
675:
672:
669:
647:
644:
641:
628:does not hold
626:
625:
614:
611:
608:
605:
602:
599:
596:
593:
590:
576:
575:
564:
561:
558:
555:
551:
548:
545:
542:
539:
515:
512:
509:
497:
496:
485:
482:
479:
476:
473:
470:
467:
464:
461:
457:
454:
451:
448:
445:
442:
439:
436:
433:
430:
403:
381:
353:
349:
325:
321:
315:
311:
307:
304:
301:
296:
292:
286:
282:
278:
275:
270:
266:
262:
257:
253:
232:
217:Main article:
214:
211:
199:transcendental
169:free variables
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
1886:
1875:
1872:
1870:
1867:
1865:
1862:
1861:
1859:
1849:
1845:
1841:
1837:
1836:
1830:
1826:
1820:
1816:
1815:
1809:
1808:
1789:
1785:
1778:
1771:
1770:Herstein 1975
1766:
1758:
1756:0-8247-6479-X
1752:
1748:
1747:Marcel Dekker
1744:
1743:
1735:
1727:
1725:0-486-65940-2
1721:
1717:
1713:
1712:
1704:
1696:
1692:
1688:
1684:
1683:
1675:
1667:
1663:
1659:
1658:
1650:
1643:
1638:
1634:
1624:
1621:
1619:
1616:
1614:
1611:
1610:
1604:
1602:
1589:
1586:
1577:
1564:
1561:
1552:
1539:
1530:
1517:
1508:
1495:
1486:
1470:
1467:
1464:
1458:
1449:
1436:
1427:
1414:
1405:
1392:
1384:
1380:
1375:
1373:
1360:
1357:
1354:
1351:
1348:
1339:
1323:
1320:
1317:
1311:
1302:
1289:
1280:
1267:
1258:
1254:
1250:
1237:
1228:
1215:
1207:
1203:
1199:
1193:
1186:As generators
1183:
1181:
1167:
1164:
1159:
1155:
1151:
1148:
1145:
1142:
1139:
1134:
1130:
1126:
1123:
1118:
1114:
1110:
1107:
1104:
1101:
1098:
1089:
1076:
1067:
1063:
1059:
1046:
1025:
1022:
1017:
1013:
1007:
1003:
999:
996:
991:
987:
983:
978:
974:
965:
952:
943:
937:
927:
925:
910:
906:
902:
899:
890:
877:
868:
853:
849:
845:
842:
818:
815:
812:
807:
803:
799:
796:
792:
789:
786:
781:
777:
773:
770:
763:
762:
761:
759:
755:
742:
733:
720:
710:
708:
695:
686:
673:
670:
667:
658:
645:
642:
639:
631:
612:
609:
606:
603:
600:
597:
594:
591:
588:
581:
580:
579:
562:
559:
556:
553:
549:
546:
543:
540:
537:
530:
529:
528:
526:
513:
510:
507:
483:
480:
477:
474:
471:
465:
459:
455:
452:
449:
446:
443:
440:
434:
428:
421:
420:
419:
416:
414:
401:
392:
379:
370:
366:
351:
347:
323:
319:
313:
309:
305:
302:
299:
294:
290:
284:
280:
276:
273:
268:
264:
260:
255:
251:
230:
220:
210:
208:
200:
193:
189:
188:indeterminate
185:
180:
178:
174:
170:
166:
162:
161:indeterminate
157:
155:
150:
148:
144:
140:
136:
132:
128:
127:indeterminate
124:
113:
110:
102:
99:December 2019
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
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32:This article
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41:Please help
36:verification
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1869:Polynomials
1642:McCoy (1960
1066:convergence
213:Polynomials
143:expressions
135:polynomials
123:mathematics
1858:Categories
1838:, Boston:
1824:047102371X
1805:References
1793:2019-12-02
1666:B0006W3EBY
1090:, such as
219:Polynomial
154:operations
69:newspapers
1817:. Wiley.
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