2719:
1368:
1208:
1724:
2095:
Any number constructible out of the integers with roots, addition, and multiplication is an algebraic integer; but not all algebraic integers are so constructible: in a naĂŻve sense, most roots of irreducible
2273:
is an algebraic number then it can be written as the ratio of an algebraic integer to a non-zero algebraic integer. In fact, the denominator can always be chosen to be a positive integer. The ratio is
1611:
1363:{\displaystyle {\begin{cases}1,\alpha ,{\dfrac {\alpha ^{2}\pm k^{2}\alpha +k^{2}}{3k}}&m\equiv \pm 1{\bmod {9}}\\1,\alpha ,{\dfrac {\alpha ^{2}}{k}}&{\text{otherwise}}\end{cases}}}
880:
1115:
987:
1429:
2405:
2377:
350:
319:
627:
413:
1159:
675:
1931:
1909:
1864:
1842:
1813:
1787:
1765:
1558:
1467:
732:
706:
596:
562:
540:
497:
466:
278:
232:
202:
50:
1068:
919:
808:
2067:
Every root of a monic polynomial whose coefficients are algebraic integers is itself an algebraic integer. In other words, the algebraic integers form a ring that is
2539:
1874:
The sum, difference and product of two algebraic integers is an algebraic integer. In general their quotient is not. Thus the algebraic integers form a
2478:
1527:
1822:
The analogy is possible because both algebraic integers and algebraic numbers are defined as roots of monic polynomials over either
439:
2446:
1727:
205:
2613:
686:
The only algebraic integers that are found in the set of rational numbers are the integers. In other words, the intersection of
2532:
1719:{\displaystyle \mathbb {Z} (\alpha )\equiv \{\sum _{i=0}^{n}\alpha ^{i}z_{i}|z_{i}\in \mathbb {Z} ,n\in \mathbb {Z} \}}
2627:
1790:
2441:
2743:
2525:
2083:
843:
2506:
1073:
945:
883:
353:
143:
2101:
2068:
1396:
2382:
2354:
327:
287:
2112:
1380:
71:
604:
2119:
1937:
1535:
372:
2574:
2408:
1576:
2718:
647:
of a ring extension. In particular, an algebraic integer is an integral element of a finite extension
2569:
235:
91:
1217:
1127:
650:
2056:
1191:
834:
159:
155:
103:
1914:
1892:
1847:
1825:
1796:
1770:
1748:
1541:
1434:
715:
689:
579:
545:
514:
471:
449:
261:
215:
176:
33:
2723:
2589:
2431:
2123:
2118:
If the monic polynomial associated with an algebraic integer has constant term 1 or â1, then the
1875:
1049:
937:
900:
789:
2748:
2704:
2695:
2686:
2584:
2474:
1936:
One may also construct explicitly the monic polynomial involved, which is generally of higher
1118:
815:
60:
2026:, which is irreducible, and is the monic equation satisfied by the product. (To see that the
2681:
2676:
2671:
2666:
2661:
2594:
2579:
2548:
2493:
2426:
2421:
2108:
2079:
2075:
1886:
1882:
1815:
is finitely generated itself); the only required change is that only non-negative powers of
1742:
1738:
1568:
1391:
644:
322:
255:
121:
111:
95:
83:
64:
2632:
2564:
2470:
1601:
838:
281:
165:. Each algebraic integer belongs to the ring of integers of some number field. A number
2127:
1177:
170:
79:
1583:
is 1, which is weaker than requiring the coefficients to be pairwise relatively prime.
2737:
2637:
2618:
2608:
2436:
1387:
1021:
823:
209:
2647:
2642:
2599:
251:
125:
17:
784:
53:
99:
1941:
599:
110:
is closed under addition, subtraction and multiplication and therefore is a
106:
is 1) whose coefficients are integers. The set of all algebraic integers
27:
Complex number that solves a monic polynomial with integer coefficients
2097:
1597:
1194:
760:
114:
87:
2497:
2122:
of that algebraic integer is also an algebraic integer, and each is a
576:
is an algebraic integer if there exists a non-zero finitely generated
246:
The following are equivalent definitions of an algebraic integer. Let
2517:
2347:
The only rational algebraic integers are the integers. Thus, if
2055:, one might use the fact that the resultant is contained in the
30:
This article is about the ring of complex numbers integral over
2521:
1302:
1356:
369:
is an algebraic integer if there exists a monic polynomial
989:
is also an algebraic integer. It satisfies the polynomial
1940:
than those of the original algebraic integers, by taking
1046:
is an integer. The full ring of integers is generated by
1076:
948:
2385:
2357:
1917:
1895:
1850:
1828:
1799:
1773:
1751:
1614:
1544:
1437:
1399:
1328:
1233:
1211:
1130:
1052:
903:
846:
792:
718:
692:
653:
607:
582:
548:
517:
474:
452:
375:
330:
290:
264:
218:
179:
36:
1530:
that has integer coefficients but is not monic, and
886:
of rational numbers. The ring of algebraic integers
1793:replaced by finite generation (using the fact that
52:. For the general notion of algebraic integer, see
2399:
2371:
1925:
1903:
1858:
1836:
1807:
1781:
1759:
1718:
1552:
1461:
1423:
1362:
1153:
1109:
1062:
981:
913:
874:
802:
726:
700:
669:
621:
590:
556:
534:
491:
460:
407:
344:
313:
272:
226:
196:
44:
2499:Algebraic Number Theory: A Computational Approach
2264:is a monic polynomial with integer coefficients.
2211:is an algebraic integer because it is a root of
1491:is another algebraic integer. A polynomial for
154:: it can also be characterised as the maximal
2533:
921:since this is a root of the monic polynomial
875:{\displaystyle K=\mathbb {Q} ({\sqrt {d}}\,)}
90:. That is, an algebraic integer is a complex
8:
1713:
1632:
1110:{\textstyle {\frac {1}{2}}(1+{\sqrt {d}}\,)}
982:{\textstyle {\frac {1}{2}}(1+{\sqrt {d}}\,)}
766:. The leading coefficient of the polynomial
2540:
2526:
2518:
2393:
2392:
2384:
2365:
2364:
2356:
2059:generated by its two input polynomials.)
1919:
1918:
1916:
1897:
1896:
1894:
1852:
1851:
1849:
1830:
1829:
1827:
1801:
1800:
1798:
1775:
1774:
1772:
1753:
1752:
1750:
1709:
1708:
1695:
1694:
1685:
1676:
1670:
1660:
1650:
1639:
1616:
1615:
1613:
1546:
1545:
1543:
1450:
1439:
1438:
1436:
1412:
1401:
1400:
1398:
1348:
1334:
1327:
1305:
1301:
1269:
1253:
1240:
1232:
1212:
1210:
1138:
1137:
1129:
1103:
1096:
1077:
1075:
1053:
1051:
975:
968:
949:
947:
904:
902:
868:
861:
854:
853:
845:
793:
791:
720:
719:
717:
694:
693:
691:
663:
662:
657:
652:
643:Algebraic integers are a special case of
615:
614:
606:
584:
583:
581:
550:
549:
547:
519:
518:
516:
476:
475:
473:
454:
453:
451:
392:
391:
374:
338:
337:
329:
298:
297:
289:
266:
265:
263:
220:
219:
217:
181:
180:
178:
38:
37:
35:
1424:{\displaystyle \mathbb {Q} (\zeta _{n})}
2457:
2400:{\displaystyle \alpha \in \mathbb {Z} }
2372:{\displaystyle \alpha \in \mathbb {Q} }
345:{\displaystyle \theta \in \mathbb {C} }
314:{\displaystyle K=\mathbb {Q} (\theta )}
2512:from the original on November 2, 2013.
1737:The proof is analogous to that of the
622:{\displaystyle M\subset \mathbb {C} }
7:
2319:with integer coefficients and where
2168:with integer coefficients and where
2107:The ring of algebraic integers is a
1600:(in the sense that is equivalent to
408:{\displaystyle f(x)\in \mathbb {Z} }
2411:for the case of a monic polynomial.
1588:Finite generation of ring extension
1390:, then the ring of integers of the
756:is not an algebraic integer unless
2469:(3rd ed.). Berlin, New York:
2130:of the ring of algebraic integers.
1124:The ring of integers of the field
25:
2407:. This is a direct result of the
2074:Again, the proof is analogous to
1995:and the polynomials satisfied by
1881:This can be shown analogously to
2717:
814:is an algebraic integer, but is
2153:is an algebraic integer, where
1944:and factoring. For example, if
438:is an algebraic integer if the
2333:is the highest-degree term of
2182:is the highest-degree term of
1677:
1626:
1620:
1575:is used in the sense that the
1456:
1443:
1418:
1405:
1154:{\displaystyle F=\mathbb {Q} }
1148:
1142:
1104:
1087:
976:
959:
869:
858:
670:{\displaystyle K/\mathbb {Q} }
529:
523:
486:
480:
402:
396:
385:
379:
308:
302:
191:
185:
1:
2351:is an algebraic integers and
1476:is an algebraic integer then
2139:is an algebraic number then
1926:{\displaystyle \mathbb {Q} }
1904:{\displaystyle \mathbb {Z} }
1859:{\displaystyle \mathbb {Q} }
1837:{\displaystyle \mathbb {Z} }
1819:are involved in the proof.
1808:{\displaystyle \mathbb {Z} }
1782:{\displaystyle \mathbb {Z} }
1760:{\displaystyle \mathbb {Q} }
1564:are algebraic integers (but
1560:, then none of the roots of
1553:{\displaystyle \mathbb {Q} }
1495:is obtained by substituting
1462:{\displaystyle \mathbb {Z} }
727:{\displaystyle \mathbb {Z} }
701:{\displaystyle \mathbb {Q} }
591:{\displaystyle \mathbb {Z} }
557:{\displaystyle \mathbb {Z} }
535:{\displaystyle \mathbb {Z} }
492:{\displaystyle \mathbb {Z} }
461:{\displaystyle \mathbb {Q} }
273:{\displaystyle \mathbb {Q} }
227:{\displaystyle \mathbb {Z} }
197:{\displaystyle \mathbb {Z} }
45:{\displaystyle \mathbb {Z} }
2628:Quadratic irrational number
2614:PisotâVijayaraghavan number
2473:. ch. 2, p. 38 and ex. 41.
1063:{\displaystyle {\sqrt {d}}}
914:{\displaystyle {\sqrt {d}}}
803:{\displaystyle {\sqrt {n}}}
511:is an algebraic integer if
2765:
2465:Marcus, Daniel A. (1977).
2111:, as a consequence of the
2071:in any of its extensions.
2003:using the resultant gives
58:
29:
2713:
2555:
1911:instead of the rationals
1734:is an algebraic integer.
810:of a nonnegative integer
354:primitive element theorem
2442:Dirichlet's unit theorem
1789:here, and the notion of
542:is a finitely generated
212:, which is to say, as a
169:is an algebraic integer
117:of the complex numbers.
59:Not to be confused with
2308:satisfies a polynomial
2157:satisfies a polynomial
2113:principal ideal theorem
2076:the corresponding proof
1883:the corresponding proof
1579:of the coefficients of
72:algebraic number theory
2724:Mathematics portal
2401:
2373:
1927:
1905:
1860:
1838:
1809:
1791:field extension degree
1783:
1761:
1720:
1655:
1554:
1501:in the polynomial for
1463:
1425:
1364:
1155:
1111:
1064:
983:
915:
876:
804:
734:. The rational number
728:
702:
671:
623:
592:
558:
536:
493:
462:
409:
346:
315:
274:
228:
198:
46:
2409:rational root theorem
2402:
2374:
2100:are not. This is the
1928:
1906:
1889:, using the integers
1861:
1839:
1810:
1784:
1762:
1721:
1635:
1604:) of the integers by
1577:highest common factor
1555:
1464:
1426:
1365:
1156:
1112:
1065:
984:
916:
877:
805:
729:
703:
672:
624:
593:
559:
537:
494:
463:
410:
347:
316:
275:
229:
199:
47:
2570:Constructible number
2383:
2355:
2126:, an element of the
2102:AbelâRuffini theorem
2084:algebraically closed
1915:
1893:
1848:
1826:
1797:
1771:
1749:
1612:
1542:
1528:primitive polynomial
1435:
1397:
1209:
1176:, has the following
1128:
1074:
1050:
946:
901:
844:
790:
716:
690:
651:
605:
580:
546:
515:
472:
450:
442:monic polynomial of
373:
328:
288:
262:
216:
177:
34:
2696:Supersilver ratio (
2687:Supergolden ratio (
1976:, then eliminating
942:, then the element
835:square-free integer
284:), in other words,
104:leading coefficient
2590:Eisenstein integer
2432:Eisenstein integer
2397:
2369:
1923:
1901:
1856:
1834:
1805:
1779:
1767:there replaced by
1757:
1739:corresponding fact
1728:finitely generated
1716:
1550:
1459:
1421:
1360:
1355:
1344:
1285:
1151:
1117:respectively. See
1107:
1060:
979:
911:
872:
800:
724:
698:
667:
619:
588:
554:
532:
489:
458:
405:
342:
311:
270:
224:
206:finitely generated
194:
42:
18:Algebraic integers
2744:Algebraic numbers
2731:
2730:
2705:Twelfth root of 2
2585:Doubling the cube
2575:Conway's constant
2560:Algebraic integer
2549:Algebraic numbers
2480:978-0-387-90279-1
2447:Fundamental units
2080:algebraic numbers
2069:integrally closed
2030:is a root of the
1887:algebraic numbers
1743:algebraic numbers
1569:algebraic numbers
1351:
1343:
1284:
1119:Quadratic integer
1101:
1085:
1058:
973:
957:
909:
866:
798:
645:integral elements
76:algebraic integer
61:algebraic element
16:(Redirected from
2756:
2722:
2721:
2699:
2690:
2682:Square root of 7
2677:Square root of 6
2672:Square root of 5
2667:Square root of 3
2662:Square root of 2
2655:
2651:
2622:
2603:
2595:Gaussian integer
2580:Cyclotomic field
2542:
2535:
2528:
2519:
2513:
2511:
2504:
2485:
2484:
2462:
2427:Gaussian integer
2422:Integral element
2406:
2404:
2403:
2398:
2396:
2378:
2376:
2375:
2370:
2368:
2350:
2343:
2332:
2318:
2307:
2303:
2301:
2286:
2272:
2263:
2252:
2233:
2232:
2210:
2192:
2181:
2167:
2156:
2152:
2138:
2090:Additional facts
2063:Integral closure
2054:
2043:
2033:
2029:
2025:
2002:
1998:
1994:
1983:
1979:
1975:
1965:
1954:
1932:
1930:
1929:
1924:
1922:
1910:
1908:
1907:
1902:
1900:
1866:, respectively.
1865:
1863:
1862:
1857:
1855:
1843:
1841:
1840:
1835:
1833:
1818:
1814:
1812:
1811:
1806:
1804:
1788:
1786:
1785:
1780:
1778:
1766:
1764:
1763:
1758:
1756:
1733:
1725:
1723:
1722:
1717:
1712:
1698:
1690:
1689:
1680:
1675:
1674:
1665:
1664:
1654:
1649:
1619:
1607:
1595:
1582:
1563:
1559:
1557:
1556:
1551:
1549:
1533:
1525:
1504:
1500:
1494:
1490:
1489:
1488:
1475:
1468:
1466:
1465:
1460:
1455:
1454:
1442:
1430:
1428:
1427:
1422:
1417:
1416:
1404:
1392:cyclotomic field
1385:
1378:
1369:
1367:
1366:
1361:
1359:
1358:
1352:
1349:
1345:
1339:
1338:
1329:
1310:
1309:
1286:
1283:
1275:
1274:
1273:
1258:
1257:
1245:
1244:
1234:
1204:
1200:
1189:
1175:
1174:
1173:
1160:
1158:
1157:
1152:
1141:
1116:
1114:
1113:
1108:
1102:
1097:
1086:
1078:
1069:
1067:
1066:
1061:
1059:
1054:
1045:
1039:
1037:
1036:
1033:
1030:
1019:
1013:
1011:
1010:
1007:
1004:
988:
986:
985:
980:
974:
969:
958:
950:
941:
930:
920:
918:
917:
912:
910:
905:
896:
890:
881:
879:
878:
873:
867:
862:
857:
832:
821:
813:
809:
807:
806:
801:
799:
794:
779:
775:
765:
759:
755:
754:
752:
751:
746:
743:
733:
731:
730:
725:
723:
711:
707:
705:
704:
699:
697:
676:
674:
673:
668:
666:
661:
638:
628:
626:
625:
620:
618:
597:
595:
594:
589:
587:
575:
563:
561:
560:
555:
553:
541:
539:
538:
533:
522:
510:
498:
496:
495:
490:
479:
467:
465:
464:
459:
457:
445:
437:
425:
414:
412:
411:
406:
395:
368:
351:
349:
348:
343:
341:
323:algebraic number
320:
318:
317:
312:
301:
282:rational numbers
279:
277:
276:
271:
269:
256:finite extension
249:
233:
231:
230:
225:
223:
203:
201:
200:
195:
184:
168:
164:
153:
149:
141:
135:
130:
122:ring of integers
109:
96:monic polynomial
65:algebraic number
51:
49:
48:
43:
41:
21:
2764:
2763:
2759:
2758:
2757:
2755:
2754:
2753:
2734:
2733:
2732:
2727:
2716:
2709:
2697:
2688:
2656:
2653:
2649:
2633:Rational number
2620:
2619:Plastic ratio (
2601:
2565:Chebyshev nodes
2551:
2546:
2516:
2509:
2502:
2492:
2488:
2481:
2471:Springer-Verlag
2464:
2463:
2459:
2455:
2418:
2381:
2380:
2353:
2352:
2348:
2334:
2328:
2320:
2309:
2305:
2300:
2292:
2285:
2277:
2275:
2268:
2254:
2250:
2231:
2226:
2225:
2224:
2212:
2206:
2194:
2183:
2177:
2169:
2158:
2154:
2148:
2140:
2134:
2092:
2065:
2045:
2035:
2031:
2027:
2004:
2000:
1996:
1985:
1981:
1977:
1967:
1956:
1945:
1913:
1912:
1891:
1890:
1872:
1846:
1845:
1824:
1823:
1816:
1795:
1794:
1769:
1768:
1747:
1746:
1731:
1730:if and only if
1681:
1666:
1656:
1610:
1609:
1605:
1602:field extension
1593:
1590:
1580:
1561:
1540:
1539:
1531:
1516:
1512:
1502:
1496:
1492:
1484:
1482:
1477:
1473:
1446:
1433:
1432:
1408:
1395:
1394:
1383:
1377:
1373:
1354:
1353:
1346:
1330:
1312:
1311:
1287:
1276:
1265:
1249:
1236:
1235:
1213:
1207:
1206:
1202:
1198:
1181:
1169:
1167:
1162:
1126:
1125:
1072:
1071:
1048:
1047:
1034:
1031:
1028:
1027:
1025:
1024:
1008:
1005:
1002:
1001:
999:
990:
944:
943:
932:
931:. Moreover, if
922:
899:
898:
895:
888:
887:
884:quadratic field
842:
841:
830:
819:
811:
788:
787:
777:
776:is the integer
767:
763:
757:
747:
744:
739:
738:
736:
735:
714:
713:
709:
688:
687:
683:
649:
648:
630:
603:
602:
578:
577:
567:
544:
543:
513:
512:
502:
470:
469:
448:
447:
443:
429:
416:
371:
370:
360:
326:
325:
286:
285:
280:, the field of
260:
259:
247:
244:
214:
213:
175:
174:
166:
162:
151:
147:
140:
133:
132:
128:
107:
68:
57:
32:
31:
28:
23:
22:
15:
12:
11:
5:
2762:
2760:
2752:
2751:
2746:
2736:
2735:
2729:
2728:
2714:
2711:
2710:
2708:
2707:
2702:
2693:
2684:
2679:
2674:
2669:
2664:
2659:
2652:
2648:Silver ratio (
2645:
2640:
2635:
2630:
2625:
2616:
2611:
2606:
2600:Golden ratio (
2597:
2592:
2587:
2582:
2577:
2572:
2567:
2562:
2556:
2553:
2552:
2547:
2545:
2544:
2537:
2530:
2522:
2515:
2514:
2494:Stein, William
2489:
2487:
2486:
2479:
2456:
2454:
2451:
2450:
2449:
2444:
2439:
2434:
2429:
2424:
2417:
2414:
2413:
2412:
2395:
2391:
2388:
2367:
2363:
2360:
2345:
2324:
2296:
2281:
2265:
2246:
2227:
2202:
2173:
2144:
2131:
2128:group of units
2116:
2105:
2091:
2088:
2064:
2061:
2034:-resultant of
1921:
1899:
1871:
1868:
1854:
1832:
1803:
1777:
1755:
1715:
1711:
1707:
1704:
1701:
1697:
1693:
1688:
1684:
1679:
1673:
1669:
1663:
1659:
1653:
1648:
1645:
1642:
1638:
1634:
1631:
1628:
1625:
1622:
1618:
1598:ring extension
1589:
1586:
1585:
1584:
1548:
1511:
1508:
1507:
1506:
1470:
1458:
1453:
1449:
1445:
1441:
1420:
1415:
1411:
1407:
1403:
1375:
1370:
1357:
1347:
1342:
1337:
1333:
1326:
1323:
1320:
1317:
1314:
1313:
1308:
1304:
1300:
1297:
1294:
1291:
1288:
1282:
1279:
1272:
1268:
1264:
1261:
1256:
1252:
1248:
1243:
1239:
1231:
1228:
1225:
1222:
1219:
1218:
1216:
1178:integral basis
1150:
1147:
1144:
1140:
1136:
1133:
1122:
1106:
1100:
1095:
1092:
1089:
1084:
1081:
1057:
978:
972:
967:
964:
961:
956:
953:
908:
891:
871:
865:
860:
856:
852:
849:
827:
824:perfect square
797:
781:
722:
696:
682:
679:
665:
660:
656:
641:
640:
617:
613:
610:
586:
565:
552:
531:
528:
525:
521:
500:
488:
485:
482:
478:
456:
427:
404:
401:
398:
394:
390:
387:
384:
381:
378:
340:
336:
333:
310:
307:
304:
300:
296:
293:
268:
243:
240:
222:
193:
190:
187:
183:
171:if and only if
136:
80:complex number
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2761:
2750:
2747:
2745:
2742:
2741:
2739:
2726:
2725:
2720:
2712:
2706:
2703:
2701:
2694:
2692:
2685:
2683:
2680:
2678:
2675:
2673:
2670:
2668:
2665:
2663:
2660:
2658:
2646:
2644:
2641:
2639:
2638:Root of unity
2636:
2634:
2631:
2629:
2626:
2624:
2617:
2615:
2612:
2610:
2609:Perron number
2607:
2605:
2598:
2596:
2593:
2591:
2588:
2586:
2583:
2581:
2578:
2576:
2573:
2571:
2568:
2566:
2563:
2561:
2558:
2557:
2554:
2550:
2543:
2538:
2536:
2531:
2529:
2524:
2523:
2520:
2508:
2501:
2500:
2495:
2491:
2490:
2482:
2476:
2472:
2468:
2467:Number Fields
2461:
2458:
2452:
2448:
2445:
2443:
2440:
2438:
2437:Root of unity
2435:
2433:
2430:
2428:
2425:
2423:
2420:
2419:
2415:
2410:
2389:
2386:
2361:
2358:
2346:
2341:
2337:
2331:
2327:
2323:
2316:
2312:
2299:
2295:
2290:
2284:
2280:
2271:
2266:
2261:
2257:
2249:
2245:
2241:
2237:
2230:
2223:
2219:
2215:
2209:
2205:
2201:
2197:
2193:. The value
2190:
2186:
2180:
2176:
2172:
2165:
2161:
2151:
2147:
2143:
2137:
2132:
2129:
2125:
2121:
2117:
2114:
2110:
2109:BĂ©zout domain
2106:
2103:
2099:
2094:
2093:
2089:
2087:
2085:
2081:
2077:
2072:
2070:
2062:
2060:
2058:
2052:
2048:
2042:
2038:
2023:
2019:
2015:
2011:
2007:
1992:
1988:
1974:
1970:
1963:
1959:
1952:
1948:
1943:
1939:
1934:
1888:
1884:
1879:
1877:
1869:
1867:
1820:
1792:
1744:
1740:
1735:
1729:
1705:
1702:
1699:
1691:
1686:
1682:
1671:
1667:
1661:
1657:
1651:
1646:
1643:
1640:
1636:
1629:
1623:
1608:, denoted by
1603:
1599:
1587:
1578:
1574:
1570:
1567:
1537:
1529:
1523:
1519:
1514:
1513:
1509:
1499:
1487:
1480:
1471:
1451:
1447:
1431:is precisely
1413:
1409:
1393:
1389:
1388:root of unity
1382:
1371:
1340:
1335:
1331:
1324:
1321:
1318:
1315:
1306:
1298:
1295:
1292:
1289:
1280:
1277:
1270:
1266:
1262:
1259:
1254:
1250:
1246:
1241:
1237:
1229:
1226:
1223:
1220:
1214:
1196:
1193:
1188:
1184:
1179:
1172:
1165:
1145:
1134:
1131:
1123:
1120:
1098:
1093:
1090:
1082:
1079:
1055:
1043:
1023:
1022:constant term
1017:
997:
993:
970:
965:
962:
954:
951:
939:
935:
929:
925:
906:
894:
885:
863:
850:
847:
840:
836:
828:
825:
817:
795:
786:
782:
774:
770:
762:
750:
742:
685:
684:
680:
678:
658:
654:
646:
637:
633:
611:
608:
601:
574:
570:
566:
526:
509:
505:
501:
483:
441:
436:
432:
428:
423:
419:
399:
388:
382:
376:
367:
363:
359:
358:
357:
355:
334:
331:
324:
305:
294:
291:
283:
257:
253:
241:
239:
237:
211:
210:abelian group
207:
188:
172:
161:
157:
145:
139:
131:, denoted by
127:
123:
118:
116:
113:
105:
101:
97:
93:
89:
85:
81:
77:
73:
66:
62:
55:
19:
2715:
2643:Salem number
2559:
2498:
2466:
2460:
2339:
2335:
2329:
2325:
2321:
2314:
2310:
2297:
2293:
2288:
2282:
2278:
2269:
2259:
2255:
2247:
2243:
2239:
2235:
2228:
2221:
2217:
2213:
2207:
2203:
2199:
2195:
2188:
2184:
2178:
2174:
2170:
2163:
2159:
2149:
2145:
2141:
2135:
2073:
2066:
2050:
2046:
2040:
2036:
2021:
2017:
2013:
2009:
2005:
1990:
1986:
1972:
1968:
1961:
1957:
1950:
1946:
1935:
1880:
1873:
1821:
1736:
1591:
1572:
1565:
1521:
1517:
1497:
1485:
1478:
1186:
1182:
1170:
1163:
1041:
1015:
995:
991:
933:
927:
923:
892:
772:
768:
748:
740:
642:
635:
631:
572:
568:
507:
503:
434:
430:
421:
417:
365:
361:
252:number field
245:
144:intersection
137:
126:number field
119:
75:
69:
1536:irreducible
1510:Non-example
1192:square-free
785:square root
712:is exactly
242:Definitions
112:commutative
54:Integrality
2738:Categories
2453:References
2120:reciprocal
1942:resultants
1741:regarding
1180:, writing
1020:where the
816:irrational
629:such that
415:such that
100:polynomial
2390:∈
2387:α
2362:∈
2359:α
1706:∈
1692:∈
1658:α
1637:∑
1630:≡
1624:α
1573:primitive
1448:ζ
1410:ζ
1381:primitive
1350:otherwise
1332:α
1322:α
1296:±
1293:≡
1260:α
1247:±
1238:α
1227:α
1197:integers
1146:α
1121:for more.
897:contains
839:extension
837:then the
612:⊂
600:submodule
527:α
389:∈
335:∈
332:θ
321:for some
306:θ
254:(i.e., a
189:α
173:the ring
142:, is the
86:over the
2749:Integers
2507:Archived
2416:See also
2304:, where
2291:/ |
2253:, where
2242: /
2098:quintics
1592:For any
1571:). Here
1190:for two
681:Examples
564:-module.
94:of some
88:integers
84:integral
82:that is
2379:, then
2234:
2024:â 1 = 0
1964:â 1 = 0
1953:â 1 = 0
1745:, with
1483:√
1195:coprime
1168:√
1038:
1026:
1012:
1000:
818:unless
761:divides
753:
737:
440:minimal
352:by the
158:of the
115:subring
2477:
2349:α
2302:|
2287:|
2276:|
2082:being
1938:degree
1817:α
1732:α
1606:α
1596:, the
1594:α
468:is in
236:module
208:as an
102:whose
2510:(PDF)
2503:(PDF)
2057:ideal
1984:from
1726:, is
1538:over
1526:is a
1379:is a
1040:(1 â
1014:(1 â
882:is a
833:is a
822:is a
446:over
424:) = 0
250:be a
160:field
156:order
124:of a
78:is a
74:, an
2475:ISBN
2220:) =
2124:unit
2078:for
2044:and
1999:and
1980:and
1966:and
1885:for
1876:ring
1870:Ring
1201:and
936:⥠1
783:The
708:and
150:and
120:The
92:root
2267:If
2133:If
2053:â 1
2012:â 4
2008:â 3
1993:= 0
1844:or
1566:are
1534:is
1515:If
1472:If
1386:th
1372:If
1303:mod
1070:or
938:mod
829:If
258:of
204:is
146:of
98:(a
70:In
63:or
2740::
2505:.
2496:.
2198:=
2086:.
2049:â
2041:xy
2039:â
2028:xy
2020:+
2016:+
1991:xy
1989:â
1973:xy
1971:=
1960:â
1955:,
1949:â
1933:.
1878:.
1481:=
1205::
1187:hk
1185:=
1166:=
1161:,
998:+
994:â
926:â
771:â
769:bx
677:.
634:â
632:αM
571:â
506:â
433:â
364:â
356:.
238:.
2700:)
2698:Ï
2691:)
2689:Ï
2657:)
2654:S
2650:ÎŽ
2623:)
2621:Ï
2604:)
2602:Ï
2541:e
2534:t
2527:v
2483:.
2394:Z
2366:Q
2344:.
2342:)
2340:x
2338:(
2336:p
2330:x
2326:n
2322:a
2317:)
2315:x
2313:(
2311:p
2306:x
2298:n
2294:a
2289:x
2283:n
2279:a
2270:x
2262:)
2260:y
2258:(
2256:q
2251:)
2248:n
2244:a
2240:y
2238:(
2236:p
2229:n
2222:a
2218:y
2216:(
2214:q
2208:x
2204:n
2200:a
2196:y
2191:)
2189:x
2187:(
2185:p
2179:x
2175:n
2171:a
2166:)
2164:x
2162:(
2160:p
2155:x
2150:x
2146:n
2142:a
2136:x
2115:.
2104:.
2051:x
2047:x
2037:z
2032:x
2022:z
2018:z
2014:z
2010:z
2006:z
2001:y
1997:x
1987:z
1982:y
1978:x
1969:z
1962:y
1958:y
1951:x
1947:x
1920:Q
1898:Z
1853:Q
1831:Z
1802:Z
1776:Z
1754:Q
1714:}
1710:Z
1703:n
1700:,
1696:Z
1687:i
1683:z
1678:|
1672:i
1668:z
1662:i
1652:n
1647:0
1644:=
1641:i
1633:{
1627:)
1621:(
1617:Z
1581:P
1562:P
1547:Q
1532:P
1524:)
1522:x
1520:(
1518:P
1505:.
1503:α
1498:x
1493:ÎČ
1486:α
1479:ÎČ
1474:α
1469:.
1457:]
1452:n
1444:[
1440:Z
1419:)
1414:n
1406:(
1402:Q
1384:n
1376:n
1374:ζ
1341:k
1336:2
1325:,
1319:,
1316:1
1307:9
1299:1
1290:m
1281:k
1278:3
1271:2
1267:k
1263:+
1255:2
1251:k
1242:2
1230:,
1224:,
1221:1
1215:{
1203:k
1199:h
1183:m
1171:m
1164:α
1149:]
1143:[
1139:Q
1135:=
1132:F
1105:)
1099:d
1094:+
1091:1
1088:(
1083:2
1080:1
1056:d
1044:)
1042:d
1035:4
1032:/
1029:1
1018:)
1016:d
1009:4
1006:/
1003:1
996:x
992:x
977:)
971:d
966:+
963:1
960:(
955:2
952:1
940:4
934:d
928:d
924:x
907:d
893:K
889:O
870:)
864:d
859:(
855:Q
851:=
848:K
831:d
826:.
820:n
812:n
796:n
780:.
778:b
773:a
764:a
758:b
749:b
745:/
741:a
721:Z
710:A
695:Q
664:Q
659:/
655:K
639:.
636:M
616:C
609:M
598:-
585:Z
573:K
569:α
551:Z
530:]
524:[
520:Z
508:K
504:α
499:.
487:]
484:x
481:[
477:Z
455:Q
444:α
435:K
431:α
426:.
422:α
420:(
418:f
403:]
400:x
397:[
393:Z
386:)
383:x
380:(
377:f
366:K
362:α
339:C
309:)
303:(
299:Q
295:=
292:K
267:Q
248:K
234:-
221:Z
192:]
186:[
182:Z
167:α
163:K
152:A
148:K
138:K
134:O
129:K
108:A
67:.
56:.
39:Z
20:)
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