2708:
1357:
1197:
1713:
2084:
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
2262:
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
1600:
1352:{\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}}}
869:
1104:
976:
1418:
2394:
2366:
339:
308:
616:
402:
1148:
664:
1920:
1898:
1853:
1831:
1802:
1776:
1754:
1547:
1456:
721:
695:
585:
551:
529:
486:
455:
267:
221:
191:
39:
1057:
908:
797:
2056:
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
2528:
1863:
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
2467:
1516:
1811:
The analogy is possible because both algebraic integers and algebraic numbers are defined as roots of monic polynomials over either
428:
2435:
1716:
194:
2602:
675:
The only algebraic integers that are found in the set of rational numbers are the integers. In other words, the intersection of
2521:
1708:{\displaystyle \mathbb {Z} (\alpha )\equiv \{\sum _{i=0}^{n}\alpha ^{i}z_{i}|z_{i}\in \mathbb {Z} ,n\in \mathbb {Z} \}}
2616:
1779:
2430:
2732:
2514:
2072:
832:
2495:
1062:
934:
872:
342:
132:
2090:
2057:
1385:
2371:
2343:
316:
276:
2101:
1369:
60:
593:
2108:
1926:
1524:
361:
2563:
2397:
1565:
2707:
636:
of a ring extension. In particular, an algebraic integer is an integral element of a finite extension
2558:
224:
80:
1206:
1116:
639:
2045:
1180:
823:
148:
144:
92:
1903:
1881:
1836:
1814:
1785:
1759:
1737:
1530:
1423:
704:
678:
568:
534:
503:
460:
438:
250:
204:
165:
22:
2712:
2578:
2420:
2112:
2107:
If the monic polynomial associated with an algebraic integer has constant term 1 or â1, then the
1864:
1038:
926:
889:
778:
2737:
2693:
2684:
2675:
2573:
2463:
1925:
One may also construct explicitly the monic polynomial involved, which is generally of higher
1107:
804:
49:
2015:, which is irreducible, and is the monic equation satisfied by the product. (To see that the
2670:
2665:
2660:
2655:
2650:
2583:
2568:
2537:
2482:
2415:
2410:
2097:
2068:
2064:
1875:
1871:
1804:
is finitely generated itself); the only required change is that only non-negative powers of
1731:
1727:
1557:
1380:
633:
311:
244:
110:
100:
84:
72:
53:
2621:
2553:
2459:
1590:
827:
270:
154:. Each algebraic integer belongs to the ring of integers of some number field. A number
2116:
1166:
159:
68:
1572:
is 1, which is weaker than requiring the coefficients to be pairwise relatively prime.
2726:
2626:
2607:
2597:
2425:
1376:
1010:
812:
198:
2636:
2631:
2588:
240:
114:
773:
42:
88:
1930:
588:
99:
is closed under addition, subtraction and multiplication and therefore is a
95:
is 1) whose coefficients are integers. The set of all algebraic integers
16:
Complex number that solves a monic polynomial with integer coefficients
2086:
1586:
1183:
749:
103:
76:
2486:
2111:
of that algebraic integer is also an algebraic integer, and each is a
565:
is an algebraic integer if there exists a non-zero finitely generated
235:
The following are equivalent definitions of an algebraic integer. Let
2506:
2336:
The only rational algebraic integers are the integers. Thus, if
2044:, one might use the fact that the resultant is contained in the
19:
This article is about the ring of complex numbers integral over
2510:
1291:
1345:
358:
is an algebraic integer if there exists a monic polynomial
978:
is also an algebraic integer. It satisfies the polynomial
1929:
than those of the original algebraic integers, by taking
1035:
is an integer. The full ring of integers is generated by
1065:
937:
2374:
2346:
1906:
1884:
1839:
1817:
1788:
1762:
1740:
1603:
1533:
1426:
1388:
1317:
1222:
1200:
1119:
1041:
892:
835:
781:
707:
681:
642:
596:
571:
537:
506:
463:
441:
364:
319:
279:
253:
207:
168:
25:
1519:
that has integer coefficients but is not monic, and
875:
of rational numbers. The ring of algebraic integers
1782:replaced by finite generation (using the fact that
41:. For the general notion of algebraic integer, see
2388:
2360:
1914:
1892:
1847:
1825:
1796:
1770:
1748:
1707:
1541:
1450:
1412:
1351:
1142:
1098:
1051:
970:
902:
863:
791:
715:
689:
658:
610:
579:
545:
523:
480:
449:
396:
333:
302:
261:
215:
185:
33:
2488:Algebraic Number Theory: A Computational Approach
2253:is a monic polynomial with integer coefficients.
2200:is an algebraic integer because it is a root of
1480:is another algebraic integer. A polynomial for
143:: it can also be characterised as the maximal
2522:
910:since this is a root of the monic polynomial
864:{\displaystyle K=\mathbb {Q} ({\sqrt {d}}\,)}
79:. That is, an algebraic integer is a complex
8:
1702:
1621:
1099:{\textstyle {\frac {1}{2}}(1+{\sqrt {d}}\,)}
971:{\textstyle {\frac {1}{2}}(1+{\sqrt {d}}\,)}
755:. The leading coefficient of the polynomial
2529:
2515:
2507:
2382:
2381:
2373:
2354:
2353:
2345:
2048:generated by its two input polynomials.)
1908:
1907:
1905:
1886:
1885:
1883:
1841:
1840:
1838:
1819:
1818:
1816:
1790:
1789:
1787:
1764:
1763:
1761:
1742:
1741:
1739:
1698:
1697:
1684:
1683:
1674:
1665:
1659:
1649:
1639:
1628:
1605:
1604:
1602:
1535:
1534:
1532:
1439:
1428:
1427:
1425:
1401:
1390:
1389:
1387:
1337:
1323:
1316:
1294:
1290:
1258:
1242:
1229:
1221:
1201:
1199:
1127:
1126:
1118:
1092:
1085:
1066:
1064:
1042:
1040:
964:
957:
938:
936:
893:
891:
857:
850:
843:
842:
834:
782:
780:
709:
708:
706:
683:
682:
680:
652:
651:
646:
641:
632:Algebraic integers are a special case of
604:
603:
595:
573:
572:
570:
539:
538:
536:
508:
507:
505:
465:
464:
462:
443:
442:
440:
381:
380:
363:
327:
326:
318:
287:
286:
278:
255:
254:
252:
209:
208:
206:
170:
169:
167:
27:
26:
24:
1413:{\displaystyle \mathbb {Q} (\zeta _{n})}
2446:
2389:{\displaystyle \alpha \in \mathbb {Z} }
2361:{\displaystyle \alpha \in \mathbb {Q} }
334:{\displaystyle \theta \in \mathbb {C} }
303:{\displaystyle K=\mathbb {Q} (\theta )}
2501:from the original on November 2, 2013.
1726:The proof is analogous to that of the
611:{\displaystyle M\subset \mathbb {C} }
7:
2308:with integer coefficients and where
2157:with integer coefficients and where
2096:The ring of algebraic integers is a
1589:(in the sense that is equivalent to
397:{\displaystyle f(x)\in \mathbb {Z} }
2400:for the case of a monic polynomial.
1577:Finite generation of ring extension
1379:, then the ring of integers of the
745:is not an algebraic integer unless
2458:(3rd ed.). Berlin, New York:
2119:of the ring of algebraic integers.
1113:The ring of integers of the field
14:
2396:. This is a direct result of the
2063:Again, the proof is analogous to
1984:and the polynomials satisfied by
1870:This can be shown analogously to
2706:
803:is an algebraic integer, but is
2142:is an algebraic integer, where
1933:and factoring. For example, if
427:is an algebraic integer if the
2322:is the highest-degree term of
2171:is the highest-degree term of
1666:
1615:
1609:
1564:is used in the sense that the
1445:
1432:
1407:
1394:
1143:{\displaystyle F=\mathbb {Q} }
1137:
1131:
1093:
1076:
965:
948:
858:
847:
659:{\displaystyle K/\mathbb {Q} }
518:
512:
475:
469:
391:
385:
374:
368:
297:
291:
180:
174:
1:
2340:is an algebraic integers and
1465:is an algebraic integer then
2128:is an algebraic number then
1915:{\displaystyle \mathbb {Q} }
1893:{\displaystyle \mathbb {Z} }
1848:{\displaystyle \mathbb {Q} }
1826:{\displaystyle \mathbb {Z} }
1808:are involved in the proof.
1797:{\displaystyle \mathbb {Z} }
1771:{\displaystyle \mathbb {Z} }
1749:{\displaystyle \mathbb {Q} }
1553:are algebraic integers (but
1549:, then none of the roots of
1542:{\displaystyle \mathbb {Q} }
1484:is obtained by substituting
1451:{\displaystyle \mathbb {Z} }
716:{\displaystyle \mathbb {Z} }
690:{\displaystyle \mathbb {Q} }
580:{\displaystyle \mathbb {Z} }
546:{\displaystyle \mathbb {Z} }
524:{\displaystyle \mathbb {Z} }
481:{\displaystyle \mathbb {Z} }
450:{\displaystyle \mathbb {Q} }
262:{\displaystyle \mathbb {Q} }
216:{\displaystyle \mathbb {Z} }
186:{\displaystyle \mathbb {Z} }
34:{\displaystyle \mathbb {Z} }
2617:Quadratic irrational number
2603:PisotâVijayaraghavan number
2462:. ch. 2, p. 38 and ex. 41.
1052:{\displaystyle {\sqrt {d}}}
903:{\displaystyle {\sqrt {d}}}
792:{\displaystyle {\sqrt {n}}}
500:is an algebraic integer if
2754:
2454:Marcus, Daniel A. (1977).
2100:, as a consequence of the
2060:in any of its extensions.
1992:using the resultant gives
47:
18:
2702:
2544:
1900:instead of the rationals
1723:is an algebraic integer.
799:of a nonnegative integer
343:primitive element theorem
2431:Dirichlet's unit theorem
1778:here, and the notion of
531:is a finitely generated
201:, which is to say, as a
158:is an algebraic integer
106:of the complex numbers.
48:Not to be confused with
2297:satisfies a polynomial
2146:satisfies a polynomial
2102:principal ideal theorem
2065:the corresponding proof
1872:the corresponding proof
1568:of the coefficients of
61:algebraic number theory
2713:Mathematics portal
2390:
2362:
1916:
1894:
1849:
1827:
1798:
1780:field extension degree
1772:
1750:
1709:
1644:
1543:
1490:in the polynomial for
1452:
1414:
1353:
1144:
1100:
1053:
972:
904:
865:
793:
723:. The rational number
717:
691:
660:
612:
581:
547:
525:
482:
451:
398:
335:
304:
263:
217:
187:
35:
2398:rational root theorem
2391:
2363:
2089:are not. This is the
1917:
1895:
1878:, using the integers
1850:
1828:
1799:
1773:
1751:
1710:
1624:
1593:) of the integers by
1566:highest common factor
1544:
1453:
1415:
1354:
1145:
1101:
1054:
973:
905:
866:
794:
718:
692:
661:
613:
582:
548:
526:
483:
452:
399:
336:
305:
264:
218:
188:
36:
2559:Constructible number
2372:
2344:
2115:, an element of the
2091:AbelâRuffini theorem
2073:algebraically closed
1904:
1882:
1837:
1815:
1786:
1760:
1738:
1601:
1531:
1517:primitive polynomial
1424:
1386:
1198:
1165:, has the following
1117:
1063:
1039:
935:
890:
833:
779:
705:
679:
640:
594:
569:
535:
504:
461:
439:
431:monic polynomial of
362:
317:
277:
251:
205:
166:
23:
2685:Supersilver ratio (
2676:Supergolden ratio (
1965:, then eliminating
931:, then the element
824:square-free integer
273:), in other words,
93:leading coefficient
2579:Eisenstein integer
2421:Eisenstein integer
2386:
2358:
1912:
1890:
1845:
1823:
1794:
1768:
1756:there replaced by
1746:
1728:corresponding fact
1717:finitely generated
1705:
1539:
1448:
1410:
1349:
1344:
1333:
1274:
1140:
1106:respectively. See
1096:
1049:
968:
900:
861:
789:
713:
687:
656:
608:
577:
543:
521:
478:
447:
394:
331:
300:
259:
213:
195:finitely generated
183:
31:
2733:Algebraic numbers
2720:
2719:
2694:Twelfth root of 2
2574:Doubling the cube
2564:Conway's constant
2549:Algebraic integer
2538:Algebraic numbers
2469:978-0-387-90279-1
2436:Fundamental units
2069:algebraic numbers
2058:integrally closed
2019:is a root of the
1876:algebraic numbers
1732:algebraic numbers
1558:algebraic numbers
1340:
1332:
1273:
1108:Quadratic integer
1090:
1074:
1047:
962:
946:
898:
855:
787:
634:integral elements
65:algebraic integer
50:algebraic element
2745:
2711:
2710:
2688:
2679:
2671:Square root of 7
2666:Square root of 6
2661:Square root of 5
2656:Square root of 3
2651:Square root of 2
2644:
2640:
2611:
2592:
2584:Gaussian integer
2569:Cyclotomic field
2531:
2524:
2517:
2508:
2502:
2500:
2493:
2474:
2473:
2451:
2416:Gaussian integer
2411:Integral element
2395:
2393:
2392:
2387:
2385:
2367:
2365:
2364:
2359:
2357:
2339:
2332:
2321:
2307:
2296:
2292:
2290:
2275:
2261:
2252:
2241:
2222:
2221:
2199:
2181:
2170:
2156:
2145:
2141:
2127:
2079:Additional facts
2052:Integral closure
2043:
2032:
2022:
2018:
2014:
1991:
1987:
1983:
1972:
1968:
1964:
1954:
1943:
1921:
1919:
1918:
1913:
1911:
1899:
1897:
1896:
1891:
1889:
1855:, respectively.
1854:
1852:
1851:
1846:
1844:
1832:
1830:
1829:
1824:
1822:
1807:
1803:
1801:
1800:
1795:
1793:
1777:
1775:
1774:
1769:
1767:
1755:
1753:
1752:
1747:
1745:
1722:
1714:
1712:
1711:
1706:
1701:
1687:
1679:
1678:
1669:
1664:
1663:
1654:
1653:
1643:
1638:
1608:
1596:
1584:
1571:
1552:
1548:
1546:
1545:
1540:
1538:
1522:
1514:
1493:
1489:
1483:
1479:
1478:
1477:
1464:
1457:
1455:
1454:
1449:
1444:
1443:
1431:
1419:
1417:
1416:
1411:
1406:
1405:
1393:
1381:cyclotomic field
1374:
1367:
1358:
1356:
1355:
1350:
1348:
1347:
1341:
1338:
1334:
1328:
1327:
1318:
1299:
1298:
1275:
1272:
1264:
1263:
1262:
1247:
1246:
1234:
1233:
1223:
1193:
1189:
1178:
1164:
1163:
1162:
1149:
1147:
1146:
1141:
1130:
1105:
1103:
1102:
1097:
1091:
1086:
1075:
1067:
1058:
1056:
1055:
1050:
1048:
1043:
1034:
1028:
1026:
1025:
1022:
1019:
1008:
1002:
1000:
999:
996:
993:
977:
975:
974:
969:
963:
958:
947:
939:
930:
919:
909:
907:
906:
901:
899:
894:
885:
879:
870:
868:
867:
862:
856:
851:
846:
821:
810:
802:
798:
796:
795:
790:
788:
783:
768:
764:
754:
748:
744:
743:
741:
740:
735:
732:
722:
720:
719:
714:
712:
700:
696:
694:
693:
688:
686:
665:
663:
662:
657:
655:
650:
627:
617:
615:
614:
609:
607:
586:
584:
583:
578:
576:
564:
552:
550:
549:
544:
542:
530:
528:
527:
522:
511:
499:
487:
485:
484:
479:
468:
456:
454:
453:
448:
446:
434:
426:
414:
403:
401:
400:
395:
384:
357:
340:
338:
337:
332:
330:
312:algebraic number
309:
307:
306:
301:
290:
271:rational numbers
268:
266:
265:
260:
258:
245:finite extension
238:
222:
220:
219:
214:
212:
192:
190:
189:
184:
173:
157:
153:
142:
138:
130:
124:
119:
111:ring of integers
98:
85:monic polynomial
54:algebraic number
40:
38:
37:
32:
30:
2753:
2752:
2748:
2747:
2746:
2744:
2743:
2742:
2723:
2722:
2721:
2716:
2705:
2698:
2686:
2677:
2645:
2642:
2638:
2622:Rational number
2609:
2608:Plastic ratio (
2590:
2554:Chebyshev nodes
2540:
2535:
2505:
2498:
2491:
2481:
2477:
2470:
2460:Springer-Verlag
2453:
2452:
2448:
2444:
2407:
2370:
2369:
2342:
2341:
2337:
2323:
2317:
2309:
2298:
2294:
2289:
2281:
2274:
2266:
2264:
2257:
2243:
2239:
2220:
2215:
2214:
2213:
2201:
2195:
2183:
2172:
2166:
2158:
2147:
2143:
2137:
2129:
2123:
2081:
2054:
2034:
2024:
2020:
2016:
1993:
1989:
1985:
1974:
1970:
1966:
1956:
1945:
1934:
1902:
1901:
1880:
1879:
1861:
1835:
1834:
1813:
1812:
1805:
1784:
1783:
1758:
1757:
1736:
1735:
1720:
1719:if and only if
1670:
1655:
1645:
1599:
1598:
1594:
1591:field extension
1582:
1579:
1569:
1550:
1529:
1528:
1520:
1505:
1501:
1491:
1485:
1481:
1473:
1471:
1466:
1462:
1435:
1422:
1421:
1397:
1384:
1383:
1372:
1366:
1362:
1343:
1342:
1335:
1319:
1301:
1300:
1276:
1265:
1254:
1238:
1225:
1224:
1202:
1196:
1195:
1191:
1187:
1170:
1158:
1156:
1151:
1115:
1114:
1061:
1060:
1037:
1036:
1023:
1020:
1017:
1016:
1014:
1013:
997:
994:
991:
990:
988:
979:
933:
932:
921:
920:. Moreover, if
911:
888:
887:
884:
877:
876:
873:quadratic field
831:
830:
819:
808:
800:
777:
776:
766:
765:is the integer
756:
752:
746:
736:
733:
728:
727:
725:
724:
703:
702:
698:
677:
676:
672:
638:
637:
619:
592:
591:
567:
566:
556:
533:
532:
502:
501:
491:
459:
458:
437:
436:
432:
418:
405:
360:
359:
349:
315:
314:
275:
274:
269:, the field of
249:
248:
236:
233:
203:
202:
164:
163:
155:
151:
140:
136:
129:
122:
121:
117:
96:
57:
46:
21:
20:
17:
12:
11:
5:
2751:
2749:
2741:
2740:
2735:
2725:
2724:
2718:
2717:
2703:
2700:
2699:
2697:
2696:
2691:
2682:
2673:
2668:
2663:
2658:
2653:
2648:
2641:
2637:Silver ratio (
2634:
2629:
2624:
2619:
2614:
2605:
2600:
2595:
2589:Golden ratio (
2586:
2581:
2576:
2571:
2566:
2561:
2556:
2551:
2545:
2542:
2541:
2536:
2534:
2533:
2526:
2519:
2511:
2504:
2503:
2483:Stein, William
2478:
2476:
2475:
2468:
2445:
2443:
2440:
2439:
2438:
2433:
2428:
2423:
2418:
2413:
2406:
2403:
2402:
2401:
2384:
2380:
2377:
2356:
2352:
2349:
2334:
2313:
2285:
2270:
2254:
2235:
2216:
2191:
2162:
2133:
2120:
2117:group of units
2105:
2094:
2080:
2077:
2053:
2050:
2023:-resultant of
1910:
1888:
1860:
1857:
1843:
1821:
1792:
1766:
1744:
1704:
1700:
1696:
1693:
1690:
1686:
1682:
1677:
1673:
1668:
1662:
1658:
1652:
1648:
1642:
1637:
1634:
1631:
1627:
1623:
1620:
1617:
1614:
1611:
1607:
1587:ring extension
1578:
1575:
1574:
1573:
1537:
1500:
1497:
1496:
1495:
1459:
1447:
1442:
1438:
1434:
1430:
1409:
1404:
1400:
1396:
1392:
1364:
1359:
1346:
1336:
1331:
1326:
1322:
1315:
1312:
1309:
1306:
1303:
1302:
1297:
1293:
1289:
1286:
1283:
1280:
1277:
1271:
1268:
1261:
1257:
1253:
1250:
1245:
1241:
1237:
1232:
1228:
1220:
1217:
1214:
1211:
1208:
1207:
1205:
1167:integral basis
1139:
1136:
1133:
1129:
1125:
1122:
1111:
1095:
1089:
1084:
1081:
1078:
1073:
1070:
1046:
967:
961:
956:
953:
950:
945:
942:
897:
880:
860:
854:
849:
845:
841:
838:
816:
813:perfect square
786:
770:
711:
685:
671:
668:
654:
649:
645:
630:
629:
606:
602:
599:
575:
554:
541:
520:
517:
514:
510:
489:
477:
474:
471:
467:
445:
416:
393:
390:
387:
383:
379:
376:
373:
370:
367:
329:
325:
322:
299:
296:
293:
289:
285:
282:
257:
232:
229:
211:
182:
179:
176:
172:
160:if and only if
125:
69:complex number
29:
15:
13:
10:
9:
6:
4:
3:
2:
2750:
2739:
2736:
2734:
2731:
2730:
2728:
2715:
2714:
2709:
2701:
2695:
2692:
2690:
2683:
2681:
2674:
2672:
2669:
2667:
2664:
2662:
2659:
2657:
2654:
2652:
2649:
2647:
2635:
2633:
2630:
2628:
2627:Root of unity
2625:
2623:
2620:
2618:
2615:
2613:
2606:
2604:
2601:
2599:
2598:Perron number
2596:
2594:
2587:
2585:
2582:
2580:
2577:
2575:
2572:
2570:
2567:
2565:
2562:
2560:
2557:
2555:
2552:
2550:
2547:
2546:
2543:
2539:
2532:
2527:
2525:
2520:
2518:
2513:
2512:
2509:
2497:
2490:
2489:
2484:
2480:
2479:
2471:
2465:
2461:
2457:
2456:Number Fields
2450:
2447:
2441:
2437:
2434:
2432:
2429:
2427:
2426:Root of unity
2424:
2422:
2419:
2417:
2414:
2412:
2409:
2408:
2404:
2399:
2378:
2375:
2350:
2347:
2335:
2330:
2326:
2320:
2316:
2312:
2305:
2301:
2288:
2284:
2279:
2273:
2269:
2260:
2255:
2250:
2246:
2238:
2234:
2230:
2226:
2219:
2212:
2208:
2204:
2198:
2194:
2190:
2186:
2182:. The value
2179:
2175:
2169:
2165:
2161:
2154:
2150:
2140:
2136:
2132:
2126:
2121:
2118:
2114:
2110:
2106:
2103:
2099:
2098:BĂ©zout domain
2095:
2092:
2088:
2083:
2082:
2078:
2076:
2074:
2070:
2066:
2061:
2059:
2051:
2049:
2047:
2041:
2037:
2031:
2027:
2012:
2008:
2004:
2000:
1996:
1981:
1977:
1963:
1959:
1952:
1948:
1941:
1937:
1932:
1928:
1923:
1877:
1873:
1868:
1866:
1858:
1856:
1809:
1781:
1733:
1729:
1724:
1718:
1694:
1691:
1688:
1680:
1675:
1671:
1660:
1656:
1650:
1646:
1640:
1635:
1632:
1629:
1625:
1618:
1612:
1597:, denoted by
1592:
1588:
1576:
1567:
1563:
1559:
1556:
1526:
1518:
1512:
1508:
1503:
1502:
1498:
1488:
1476:
1469:
1460:
1440:
1436:
1420:is precisely
1402:
1398:
1382:
1378:
1377:root of unity
1371:
1360:
1329:
1324:
1320:
1313:
1310:
1307:
1304:
1295:
1287:
1284:
1281:
1278:
1269:
1266:
1259:
1255:
1251:
1248:
1243:
1239:
1235:
1230:
1226:
1218:
1215:
1212:
1209:
1203:
1185:
1182:
1177:
1173:
1168:
1161:
1154:
1134:
1123:
1120:
1112:
1109:
1087:
1082:
1079:
1071:
1068:
1044:
1032:
1012:
1011:constant term
1006:
986:
982:
959:
954:
951:
943:
940:
928:
924:
918:
914:
895:
883:
874:
852:
839:
836:
829:
825:
817:
814:
806:
784:
775:
771:
763:
759:
751:
739:
731:
674:
673:
669:
667:
647:
643:
635:
626:
622:
600:
597:
590:
563:
559:
555:
515:
498:
494:
490:
472:
430:
425:
421:
417:
412:
408:
388:
377:
371:
365:
356:
352:
348:
347:
346:
344:
323:
320:
313:
294:
283:
280:
272:
246:
242:
230:
228:
226:
200:
199:abelian group
196:
177:
161:
150:
146:
134:
128:
120:, denoted by
116:
112:
107:
105:
102:
94:
90:
86:
82:
78:
74:
70:
66:
62:
55:
51:
44:
2704:
2632:Salem number
2548:
2487:
2455:
2449:
2328:
2324:
2318:
2314:
2310:
2303:
2299:
2286:
2282:
2277:
2271:
2267:
2258:
2248:
2244:
2236:
2232:
2228:
2224:
2217:
2210:
2206:
2202:
2196:
2192:
2188:
2184:
2177:
2173:
2167:
2163:
2159:
2152:
2148:
2138:
2134:
2130:
2124:
2062:
2055:
2039:
2035:
2029:
2025:
2010:
2006:
2002:
1998:
1994:
1979:
1975:
1961:
1957:
1950:
1946:
1939:
1935:
1924:
1869:
1862:
1810:
1725:
1580:
1561:
1554:
1510:
1506:
1486:
1474:
1467:
1175:
1171:
1159:
1152:
1030:
1004:
984:
980:
922:
916:
912:
881:
761:
757:
737:
729:
631:
624:
620:
561:
557:
496:
492:
423:
419:
410:
406:
354:
350:
241:number field
234:
133:intersection
126:
115:number field
108:
64:
58:
1525:irreducible
1499:Non-example
1181:square-free
774:square root
701:is exactly
231:Definitions
101:commutative
43:Integrality
2727:Categories
2442:References
2109:reciprocal
1931:resultants
1730:regarding
1169:, writing
1009:where the
805:irrational
618:such that
404:such that
89:polynomial
2379:∈
2376:α
2351:∈
2348:α
1695:∈
1681:∈
1647:α
1626:∑
1619:≡
1613:α
1562:primitive
1437:ζ
1399:ζ
1370:primitive
1339:otherwise
1321:α
1311:α
1285:±
1282:≡
1249:α
1236:±
1227:α
1216:α
1186:integers
1135:α
1110:for more.
886:contains
828:extension
826:then the
601:⊂
589:submodule
516:α
378:∈
324:∈
321:θ
310:for some
295:θ
243:(i.e., a
178:α
162:the ring
131:, is the
75:over the
2738:Integers
2496:Archived
2405:See also
2293:, where
2280:/ |
2242:, where
2231: /
2087:quintics
1581:For any
1560:). Here
1179:for two
670:Examples
553:-module.
83:of some
77:integers
73:integral
71:that is
2368:, then
2223:
2013:â 1 = 0
1953:â 1 = 0
1942:â 1 = 0
1734:, with
1472:√
1184:coprime
1157:√
1027:
1015:
1001:
989:
807:unless
750:divides
742:
726:
429:minimal
341:by the
147:of the
104:subring
2466:
2338:α
2291:|
2276:|
2265:|
2071:being
1927:degree
1806:α
1721:α
1595:α
1585:, the
1583:α
457:is in
225:module
197:as an
91:whose
2499:(PDF)
2492:(PDF)
2046:ideal
1973:from
1715:, is
1527:over
1515:is a
1368:is a
1029:(1 â
1003:(1 â
871:is a
822:is a
811:is a
435:over
413:) = 0
239:be a
149:field
145:order
113:of a
67:is a
63:, an
2464:ISBN
2209:) =
2113:unit
2067:for
2033:and
1988:and
1969:and
1955:and
1874:for
1865:ring
1859:Ring
1190:and
925:⥠1
772:The
697:and
139:and
109:The
81:root
2256:If
2122:If
2042:â 1
2001:â 4
1997:â 3
1982:= 0
1833:or
1555:are
1523:is
1504:If
1461:If
1375:th
1361:If
1292:mod
1059:or
927:mod
818:If
247:of
193:is
135:of
87:(a
59:In
52:or
2729::
2494:.
2485:.
2187:=
2075:.
2038:â
2030:xy
2028:â
2017:xy
2009:+
2005:+
1980:xy
1978:â
1962:xy
1960:=
1949:â
1944:,
1938:â
1922:.
1867:.
1470:=
1194::
1176:hk
1174:=
1155:=
1150:,
987:+
983:â
915:â
760:â
758:bx
666:.
623:â
621:αM
560:â
495:â
422:â
353:â
345:.
227:.
2689:)
2687:Ï
2680:)
2678:Ï
2646:)
2643:S
2639:ÎŽ
2612:)
2610:Ï
2593:)
2591:Ï
2530:e
2523:t
2516:v
2472:.
2383:Z
2355:Q
2333:.
2331:)
2329:x
2327:(
2325:p
2319:x
2315:n
2311:a
2306:)
2304:x
2302:(
2300:p
2295:x
2287:n
2283:a
2278:x
2272:n
2268:a
2259:x
2251:)
2249:y
2247:(
2245:q
2240:)
2237:n
2233:a
2229:y
2227:(
2225:p
2218:n
2211:a
2207:y
2205:(
2203:q
2197:x
2193:n
2189:a
2185:y
2180:)
2178:x
2176:(
2174:p
2168:x
2164:n
2160:a
2155:)
2153:x
2151:(
2149:p
2144:x
2139:x
2135:n
2131:a
2125:x
2104:.
2093:.
2040:x
2036:x
2026:z
2021:x
2011:z
2007:z
2003:z
1999:z
1995:z
1990:y
1986:x
1976:z
1971:y
1967:x
1958:z
1951:y
1947:y
1940:x
1936:x
1909:Q
1887:Z
1842:Q
1820:Z
1791:Z
1765:Z
1743:Q
1703:}
1699:Z
1692:n
1689:,
1685:Z
1676:i
1672:z
1667:|
1661:i
1657:z
1651:i
1641:n
1636:0
1633:=
1630:i
1622:{
1616:)
1610:(
1606:Z
1570:P
1551:P
1536:Q
1521:P
1513:)
1511:x
1509:(
1507:P
1494:.
1492:α
1487:x
1482:ÎČ
1475:α
1468:ÎČ
1463:α
1458:.
1446:]
1441:n
1433:[
1429:Z
1408:)
1403:n
1395:(
1391:Q
1373:n
1365:n
1363:ζ
1330:k
1325:2
1314:,
1308:,
1305:1
1296:9
1288:1
1279:m
1270:k
1267:3
1260:2
1256:k
1252:+
1244:2
1240:k
1231:2
1219:,
1213:,
1210:1
1204:{
1192:k
1188:h
1172:m
1160:m
1153:α
1138:]
1132:[
1128:Q
1124:=
1121:F
1094:)
1088:d
1083:+
1080:1
1077:(
1072:2
1069:1
1045:d
1033:)
1031:d
1024:4
1021:/
1018:1
1007:)
1005:d
998:4
995:/
992:1
985:x
981:x
966:)
960:d
955:+
952:1
949:(
944:2
941:1
929:4
923:d
917:d
913:x
896:d
882:K
878:O
859:)
853:d
848:(
844:Q
840:=
837:K
820:d
815:.
809:n
801:n
785:n
769:.
767:b
762:a
753:a
747:b
738:b
734:/
730:a
710:Z
699:A
684:Q
653:Q
648:/
644:K
628:.
625:M
605:C
598:M
587:-
574:Z
562:K
558:α
540:Z
519:]
513:[
509:Z
497:K
493:α
488:.
476:]
473:x
470:[
466:Z
444:Q
433:α
424:K
420:α
415:.
411:α
409:(
407:f
392:]
389:x
386:[
382:Z
375:)
372:x
369:(
366:f
355:K
351:α
328:C
298:)
292:(
288:Q
284:=
281:K
256:Q
237:K
223:-
210:Z
181:]
175:[
171:Z
156:α
152:K
141:A
137:K
127:K
123:O
118:K
97:A
56:.
45:.
28:Z
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