2349:
is the completion of an algebraic number field, its ring of integers is the completion of the latter's ring of integers. The ring of integers of an algebraic number field may be characterised as the elements which are integers in every non-archimedean completion.
1630:
1540:
2253:
2112:
1736:
807:
1788:
1971:
1028:
1377:
244:
536:
2146:
2006:
1819:
1661:
865:
489:
452:
1160:
2444:
1195:
1121:
1084:
1050:
991:
935:
198:
966:
834:
1925:
913:
889:
711:
683:
641:
1458:
1557:
2731:
2649:
2588:
1482:
98:
2691:
2663:
2396:
2057:
634:
586:
2723:
2176:
2784:
2312:
2289:
2063:
1679:
2161:
731:
627:
2446:
of "ordinary" integers, the prototypical object for all those rings. It is a consequence of the ambiguity of the word "
2325:
1421:
344:
2293:
1747:
2683:
2655:
1934:
996:
104:
1315:
208:
1230:
1166:, consisting of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers,
1128:
579:
503:
382:
332:
2677:
1061:
391:
84:
1256:
663:
548:
399:
350:
131:
2117:
1980:
1793:
1635:
1398:
839:
465:
428:
2779:
1239:
1134:
714:
272:
146:
2263:
2157:
1928:
1205:
1053:
554:
362:
313:
258:
152:
138:
66:
34:
2427:
1169:
1095:
1067:
1033:
974:
918:
181:
2274:
2031:
686:
567:
125:
53:
2709:
1447:
A useful tool for computing the integral closure of the ring of integers in an algebraic field
2727:
2687:
2659:
2584:
2009:
1163:
690:
608:
405:
170:
111:
2745:
2697:
2673:
2401:
2297:
1867:
1198:
1090:
892:
718:
614:
600:
356:
319:
119:
92:
78:
2741:
944:
812:
2749:
2737:
2701:
2259:
1974:
1209:
1057:
376:
326:
164:
494:
2538:
2354:
2301:
1910:
1124:
898:
874:
696:
668:
420:
2773:
2757:
2371:
1848:
561:
457:
72:
1834:
1289:
593:
368:
264:
2267:
2170:, the element 6 has two essentially different factorizations into irreducibles:
1243:
725:
655:
573:
284:
158:
40:
338:
1664:
298:
203:
1625:{\displaystyle d=\Delta _{K/\mathbb {Q} }(\alpha _{1},\ldots ,\alpha _{n})}
17:
2447:
292:
278:
1204:
The ring of integers of an algebraic number field is the unique maximal
938:
868:
722:
176:
60:
2682:. London Mathematical Society Student Texts. Vol. 3. Cambridge:
1535:{\displaystyle \alpha _{1},\ldots ,\alpha _{n}\in {\mathcal {O}}_{K}}
2343:; this is a ring because of the strong triangle inequality. If
2156:
In a ring of integers, every element has a factorization into
1131:
are integers. It is the ring of integers in the number field
2248:{\displaystyle 6=2\cdot 3=(1+{\sqrt {-5}})(1-{\sqrt {-5}}).}
1987:
1800:
1642:
1521:
846:
2107:{\displaystyle a+b{\sqrt {d}}\in \mathbf {Q} ({\sqrt {d}})}
1086:
are often called the "rational integers" because of this.
2493:
2491:
2310:. A set of torsion-free generators is called a set of
2508:
2506:
2430:
2179:
2120:
2066:
1983:
1937:
1913:
1796:
1750:
1682:
1638:
1560:
1485:
1318:
1172:
1137:
1098:
1070:
1036:
999:
977:
947:
921:
901:
877:
842:
815:
734:
699:
671:
506:
468:
431:
211:
184:
2424:, without specifying the field, refers to the ring
2404:– gives a technique for computing integral closures
1731:{\displaystyle \alpha _{1}/d,\ldots ,\alpha _{n}/d}
993:is the simplest possible ring of integers. Namely,
2438:
2247:
2140:
2106:
2000:
1965:
1919:
1813:
1782:
1730:
1655:
1624:
1534:
1371:
1189:
1154:
1115:
1078:
1044:
1022:
985:
960:
929:
907:
883:
859:
828:
802:{\displaystyle x^{n}+c_{n-1}x^{n-1}+\cdots +c_{0}}
801:
705:
677:
530:
483:
446:
238:
192:
1783:{\displaystyle \alpha _{1},\ldots ,\alpha _{n}}
2719:Grundlehren der mathematischen Wissenschaften
2482:
2470:
2160:, but the ring need not have the property of
1966:{\displaystyle K=\mathbb {Q} ({\sqrt {d}}\,)}
1023:{\displaystyle \mathbb {Z} =O_{\mathbb {Q} }}
635:
8:
2717:
1372:{\displaystyle x=\sum _{i=1}^{n}a_{i}b_{i},}
239:{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }
2648:Alaca, Saban; Williams, Kenneth S. (2003).
642:
628:
29:
2432:
2431:
2429:
2229:
2207:
2178:
2133:
2119:
2094:
2086:
2076:
2065:
1992:
1986:
1985:
1982:
1959:
1952:
1945:
1944:
1936:
1912:
1805:
1799:
1798:
1795:
1774:
1755:
1749:
1720:
1714:
1693:
1687:
1681:
1647:
1641:
1640:
1637:
1613:
1594:
1581:
1580:
1575:
1571:
1559:
1526:
1520:
1519:
1509:
1490:
1484:
1360:
1350:
1340:
1329:
1317:
1174:
1173:
1171:
1139:
1138:
1136:
1100:
1099:
1097:
1089:The next simplest example is the ring of
1072:
1071:
1069:
1038:
1037:
1035:
1014:
1013:
1012:
1001:
1000:
998:
979:
978:
976:
952:
946:
923:
922:
920:
900:
876:
851:
845:
844:
841:
820:
814:
793:
768:
752:
739:
733:
698:
670:
531:{\displaystyle \mathbb {Z} (p^{\infty })}
519:
508:
507:
505:
475:
471:
470:
467:
438:
434:
433:
430:
232:
231:
223:
219:
218:
210:
186:
185:
183:
2627:
2497:
2463:
2414:
2164:: for example, in the ring of integers
32:
2615:
2603:
2566:
2554:
2512:
2324:One defines the ring of integers of a
2262:, and so has unique factorization of
2056:. This can be found by computing the
7:
2651:Introductory Algebraic Number Theory
2532:
2530:
99:Free product of associative algebras
2141:{\displaystyle a,b\in \mathbf {Q} }
2012:and its integral basis is given by
2001:{\displaystyle {\mathcal {O}}_{K}}
1814:{\displaystyle {\mathcal {O}}_{K}}
1656:{\displaystyle {\mathcal {O}}_{K}}
1568:
860:{\displaystyle {\mathcal {O}}_{K}}
520:
25:
2397:Minimal polynomial (field theory)
587:Noncommutative algebraic geometry
2370:are the ring of integers of the
2290:finitely generated abelian group
2134:
2087:
809:. This ring is often denoted by
484:{\displaystyle \mathbb {Q} _{p}}
447:{\displaystyle \mathbb {Z} _{p}}
2258:A ring of integers is always a
1309:can be uniquely represented as
1155:{\displaystyle \mathbb {Q} (i)}
2583:. Prentice Hall. p. 360.
2333:as the set of all elements of
2239:
2220:
2217:
2198:
2101:
2091:
1960:
1949:
1619:
1587:
1184:
1178:
1149:
1143:
1110:
1104:
525:
512:
1:
2473:, p. 110, Defs. 6.1.2-3.
1208:in the field. It is always a
2485:, p. 74, Defs. 4.1.1-2.
2439:{\displaystyle \mathbb {Z} }
1870:, then an integral basis of
1790:forms an integral basis for
1296:such that each element
1190:{\displaystyle \mathbb {Z} }
1116:{\displaystyle \mathbb {Z} }
1079:{\displaystyle \mathbb {Z} }
1045:{\displaystyle \mathbb {Q} }
986:{\displaystyle \mathbb {Z} }
930:{\displaystyle \mathbb {Z} }
713:. An algebraic integer is a
193:{\displaystyle \mathbb {Z} }
2326:non-archimedean local field
345:Unique factorization domain
2801:
2714:Algebraische Zahlentheorie
2684:Cambridge University Press
2656:Cambridge University Press
105:Tensor product of algebras
2722:. Vol. 322. Berlin:
2539:"Algebraic Number Theory"
2483:Alaca & Williams 2003
2471:Alaca & Williams 2003
1251:-module, and thus has an
2294:Dirichlet's unit theorem
2152:Multiplicative structure
2060:of an arbitrary element
1420:-module is equal to the
1129:real and imaginary parts
383:Formal power series ring
333:Integrally closed domain
2785:Algebraic number theory
2762:Algebraic number theory
2579:Artin, Michael (2011).
1062:algebraic number theory
392:Algebraic number theory
85:Total ring of fractions
2718:
2450:" in abstract algebra.
2440:
2277:of a ring of integers
2249:
2142:
2108:
2002:
1967:
1921:
1815:
1784:
1732:
1657:
1626:
1536:
1373:
1345:
1191:
1156:
1117:
1080:
1046:
1024:
987:
962:
931:
909:
885:
861:
830:
803:
707:
679:
664:algebraic number field
549:Noncommutative algebra
532:
485:
448:
400:Algebraic number field
351:Principal ideal domain
240:
194:
132:Frobenius endomorphism
27:Algebraic construction
2524:Cassels (1986) p. 193
2441:
2250:
2143:
2109:
2003:
1973:is the corresponding
1968:
1922:
1866:is the corresponding
1825:Cyclotomic extensions
1816:
1785:
1744:is square-free, then
1733:
1658:
1627:
1537:
1374:
1325:
1220:The ring of integers
1192:
1157:
1118:
1081:
1047:
1025:
988:
971:The ring of integers
963:
961:{\displaystyle O_{K}}
932:
910:
886:
862:
831:
829:{\displaystyle O_{K}}
804:
708:
680:
533:
486:
449:
241:
195:
2428:
2422:The ring of integers
2339:with absolute value
2177:
2162:unique factorization
2158:irreducible elements
2118:
2064:
1981:
1935:
1911:
1903:Quadratic extensions
1794:
1748:
1680:
1636:
1558:
1483:
1316:
1170:
1135:
1096:
1068:
1034:
997:
975:
945:
919:
899:
875:
840:
813:
732:
697:
669:
555:Noncommutative rings
504:
466:
429:
273:Non-associative ring
209:
182:
139:Algebraic structures
1929:square-free integer
1242:. Indeed, it is a
314:Commutative algebra
153:Associative algebra
35:Algebraic structure
2764:. Hermann/Kershaw.
2436:
2245:
2138:
2104:
2058:minimal polynomial
2010:quadratic integers
1998:
1963:
1917:
1811:
1780:
1728:
1653:
1622:
1532:
1443:Computational tool
1369:
1231:finitely-generated
1187:
1164:Gaussian rationals
1152:
1113:
1076:
1060:. And indeed, in
1042:
1020:
983:
958:
927:
905:
881:
857:
826:
799:
703:
691:algebraic integers
675:
568:Semiprimitive ring
528:
481:
444:
252:Related structures
236:
190:
126:Inner automorphism
112:Ring homomorphisms
2733:978-3-540-65399-8
2618:, pp. 59–62.
2590:978-0-13-241377-0
2544:. pp. 33–35.
2353:For example, the
2313:fundamental units
2237:
2215:
2099:
2081:
1957:
1920:{\displaystyle d}
1091:Gaussian integers
908:{\displaystyle K}
884:{\displaystyle K}
706:{\displaystyle K}
678:{\displaystyle K}
652:
651:
609:Geometric algebra
320:Commutative rings
171:Category of rings
16:(Redirected from
2792:
2765:
2753:
2721:
2710:Neukirch, Jürgen
2705:
2669:
2631:
2625:
2619:
2613:
2607:
2601:
2595:
2594:
2576:
2570:
2564:
2558:
2552:
2546:
2545:
2543:
2534:
2525:
2522:
2516:
2510:
2501:
2495:
2486:
2480:
2474:
2468:
2451:
2445:
2443:
2442:
2437:
2435:
2419:
2402:Integral closure
2386:
2374:
2369:
2357:
2348:
2342:
2338:
2332:
2309:
2300:consists of the
2298:torsion subgroup
2287:
2254:
2252:
2251:
2246:
2238:
2230:
2216:
2208:
2169:
2147:
2145:
2144:
2139:
2137:
2113:
2111:
2110:
2105:
2100:
2095:
2090:
2082:
2077:
2055:
2048:
2046:
2045:
2035:
2024:
2022:
2021:
2007:
2005:
2004:
1999:
1997:
1996:
1991:
1990:
1972:
1970:
1969:
1964:
1958:
1953:
1948:
1926:
1924:
1923:
1918:
1898:
1882:
1868:cyclotomic field
1865:
1846:
1842:
1832:
1820:
1818:
1817:
1812:
1810:
1809:
1804:
1803:
1789:
1787:
1786:
1781:
1779:
1778:
1760:
1759:
1743:
1737:
1735:
1734:
1729:
1724:
1719:
1718:
1697:
1692:
1691:
1675:
1673:
1662:
1660:
1659:
1654:
1652:
1651:
1646:
1645:
1631:
1629:
1628:
1623:
1618:
1617:
1599:
1598:
1586:
1585:
1584:
1579:
1553:
1547:
1542:form a basis of
1541:
1539:
1538:
1533:
1531:
1530:
1525:
1524:
1514:
1513:
1495:
1494:
1478:
1472:
1466:
1456:
1433:
1427:
1419:
1413:
1404:
1396:
1378:
1376:
1375:
1370:
1365:
1364:
1355:
1354:
1344:
1339:
1308:
1299:
1295:
1287:
1281:
1250:
1237:
1228:
1199:Euclidean domain
1196:
1194:
1193:
1188:
1177:
1161:
1159:
1158:
1153:
1142:
1123:, consisting of
1122:
1120:
1119:
1114:
1103:
1085:
1083:
1082:
1077:
1075:
1064:the elements of
1058:rational numbers
1051:
1049:
1048:
1043:
1041:
1029:
1027:
1026:
1021:
1019:
1018:
1017:
1004:
992:
990:
989:
984:
982:
967:
965:
964:
959:
957:
956:
936:
934:
933:
928:
926:
914:
912:
911:
906:
893:integral element
890:
888:
887:
882:
866:
864:
863:
858:
856:
855:
850:
849:
835:
833:
832:
827:
825:
824:
808:
806:
805:
800:
798:
797:
779:
778:
763:
762:
744:
743:
719:monic polynomial
712:
710:
709:
704:
684:
682:
681:
676:
660:ring of integers
644:
637:
630:
615:Operator algebra
601:Clifford algebra
537:
535:
534:
529:
524:
523:
511:
490:
488:
487:
482:
480:
479:
474:
453:
451:
450:
445:
443:
442:
437:
415:Ring of integers
409:
406:Integers modulo
357:Euclidean domain
245:
243:
242:
237:
235:
227:
222:
199:
197:
196:
191:
189:
93:Product of rings
79:Fractional ideal
38:
30:
21:
2800:
2799:
2795:
2794:
2793:
2791:
2790:
2789:
2770:
2769:
2768:
2756:
2734:
2724:Springer-Verlag
2708:
2694:
2674:Cassels, J.W.S.
2672:
2666:
2647:
2643:
2637:
2635:
2634:
2626:
2622:
2614:
2610:
2602:
2598:
2591:
2578:
2577:
2573:
2565:
2561:
2553:
2549:
2541:
2536:
2535:
2528:
2523:
2519:
2511:
2504:
2496:
2489:
2481:
2477:
2469:
2465:
2460:
2455:
2454:
2426:
2425:
2420:
2416:
2411:
2393:
2385:
2377:
2372:
2368:
2360:
2355:
2344:
2340:
2334:
2328:
2322:
2305:
2286:
2278:
2260:Dedekind domain
2175:
2174:
2165:
2154:
2116:
2115:
2062:
2061:
2050:
2041:
2039:
2037:
2026:
2017:
2015:
2013:
1984:
1979:
1978:
1975:quadratic field
1933:
1932:
1909:
1908:
1905:
1884:
1877:
1871:
1852:
1844:
1838:
1830:
1827:
1797:
1792:
1791:
1770:
1751:
1746:
1745:
1739:
1738:. In fact, if
1710:
1683:
1678:
1677:
1669:
1668:
1639:
1634:
1633:
1609:
1590:
1567:
1556:
1555:
1549:
1543:
1518:
1505:
1486:
1481:
1480:
1474:
1468:
1462:
1448:
1445:
1440:
1429:
1425:
1415:
1412:
1406:
1402:
1391:
1383:
1356:
1346:
1314:
1313:
1307:
1301:
1297:
1293:
1283:
1280:
1274:
1265:
1259:
1246:
1233:
1227:
1221:
1218:
1210:Dedekind domain
1168:
1167:
1133:
1132:
1125:complex numbers
1094:
1093:
1066:
1065:
1032:
1031:
1008:
995:
994:
973:
972:
948:
943:
942:
917:
916:
897:
896:
873:
872:
843:
838:
837:
816:
811:
810:
789:
764:
748:
735:
730:
729:
695:
694:
667:
666:
648:
619:
618:
551:
541:
540:
515:
502:
501:
469:
464:
463:
432:
427:
426:
407:
377:Polynomial ring
327:Integral domain
316:
306:
305:
207:
206:
180:
179:
165:Involutive ring
50:
39:
33:
28:
23:
22:
15:
12:
11:
5:
2798:
2796:
2788:
2787:
2782:
2772:
2771:
2767:
2766:
2758:Samuel, Pierre
2754:
2732:
2706:
2692:
2670:
2664:
2644:
2642:
2639:
2633:
2632:
2620:
2608:
2596:
2589:
2571:
2559:
2547:
2526:
2517:
2502:
2500:, p. 192.
2487:
2475:
2462:
2461:
2459:
2456:
2453:
2452:
2434:
2413:
2412:
2410:
2407:
2406:
2405:
2399:
2392:
2389:
2381:
2364:
2358:-adic integers
2321:
2320:Generalization
2318:
2302:roots of unity
2282:
2256:
2255:
2244:
2241:
2236:
2233:
2228:
2225:
2222:
2219:
2214:
2211:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2153:
2150:
2136:
2132:
2129:
2126:
2123:
2103:
2098:
2093:
2089:
2085:
2080:
2075:
2072:
2069:
2054:≡ 2, 3 (mod 4)
1995:
1989:
1962:
1956:
1951:
1947:
1943:
1940:
1916:
1904:
1901:
1873:
1826:
1823:
1808:
1802:
1777:
1773:
1769:
1766:
1763:
1758:
1754:
1727:
1723:
1717:
1713:
1709:
1706:
1703:
1700:
1696:
1690:
1686:
1650:
1644:
1621:
1616:
1612:
1608:
1605:
1602:
1597:
1593:
1589:
1583:
1578:
1574:
1570:
1566:
1563:
1529:
1523:
1517:
1512:
1508:
1504:
1501:
1498:
1493:
1489:
1444:
1441:
1439:
1436:
1408:
1387:
1380:
1379:
1368:
1363:
1359:
1353:
1349:
1343:
1338:
1335:
1332:
1328:
1324:
1321:
1303:
1276:
1270:
1263:
1253:integral basis
1223:
1217:
1214:
1186:
1183:
1180:
1176:
1151:
1148:
1145:
1141:
1112:
1109:
1106:
1102:
1074:
1040:
1016:
1011:
1007:
1003:
981:
955:
951:
925:
904:
880:
854:
848:
823:
819:
796:
792:
788:
785:
782:
777:
774:
771:
767:
761:
758:
755:
751:
747:
742:
738:
702:
674:
650:
649:
647:
646:
639:
632:
624:
621:
620:
612:
611:
583:
582:
576:
570:
564:
552:
547:
546:
543:
542:
539:
538:
527:
522:
518:
514:
510:
491:
478:
473:
454:
441:
436:
424:-adic integers
417:
411:
402:
388:
387:
386:
385:
379:
373:
372:
371:
359:
353:
347:
341:
335:
317:
312:
311:
308:
307:
304:
303:
302:
301:
289:
288:
287:
281:
269:
268:
267:
249:
248:
247:
246:
234:
230:
226:
221:
217:
214:
200:
188:
167:
161:
155:
149:
135:
134:
128:
122:
108:
107:
101:
95:
89:
88:
87:
81:
69:
63:
51:
49:Basic concepts
48:
47:
44:
43:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2797:
2786:
2783:
2781:
2778:
2777:
2775:
2763:
2759:
2755:
2751:
2747:
2743:
2739:
2735:
2729:
2725:
2720:
2715:
2711:
2707:
2703:
2699:
2695:
2693:0-521-31525-5
2689:
2685:
2681:
2680:
2675:
2671:
2667:
2665:9780511791260
2661:
2657:
2653:
2652:
2646:
2645:
2640:
2638:
2630:, p. 41.
2629:
2624:
2621:
2617:
2612:
2609:
2606:, p. 50.
2605:
2600:
2597:
2592:
2586:
2582:
2575:
2572:
2569:, p. 35.
2568:
2563:
2560:
2557:, p. 43.
2556:
2551:
2548:
2540:
2533:
2531:
2527:
2521:
2518:
2515:, p. 49.
2514:
2509:
2507:
2503:
2499:
2494:
2492:
2488:
2484:
2479:
2476:
2472:
2467:
2464:
2457:
2449:
2423:
2418:
2415:
2408:
2403:
2400:
2398:
2395:
2394:
2390:
2388:
2384:
2380:
2376:
2375:-adic numbers
2367:
2363:
2359:
2351:
2347:
2337:
2331:
2327:
2319:
2317:
2315:
2314:
2308:
2303:
2299:
2295:
2291:
2285:
2281:
2276:
2271:
2269:
2265:
2261:
2242:
2234:
2231:
2226:
2223:
2212:
2209:
2204:
2201:
2195:
2192:
2189:
2186:
2183:
2180:
2173:
2172:
2171:
2168:
2163:
2159:
2151:
2149:
2130:
2127:
2124:
2121:
2096:
2083:
2078:
2073:
2070:
2067:
2059:
2053:
2044:
2033:
2029:
2020:
2011:
2008:is a ring of
1993:
1976:
1954:
1941:
1938:
1930:
1914:
1902:
1900:
1896:
1892:
1888:
1881:
1876:
1869:
1863:
1859:
1855:
1850:
1849:root of unity
1841:
1836:
1824:
1822:
1806:
1775:
1771:
1767:
1764:
1761:
1756:
1752:
1742:
1725:
1721:
1715:
1711:
1707:
1704:
1701:
1698:
1694:
1688:
1684:
1672:
1666:
1648:
1614:
1610:
1606:
1603:
1600:
1595:
1591:
1576:
1572:
1564:
1561:
1552:
1546:
1527:
1515:
1510:
1506:
1502:
1499:
1496:
1491:
1487:
1477:
1471:
1467:is of degree
1465:
1460:
1455:
1451:
1442:
1437:
1435:
1432:
1423:
1418:
1411:
1400:
1395:
1390:
1386:
1366:
1361:
1357:
1351:
1347:
1341:
1336:
1333:
1330:
1326:
1322:
1319:
1312:
1311:
1310:
1306:
1291:
1286:
1279:
1273:
1269:
1262:
1258:
1254:
1249:
1245:
1241:
1236:
1232:
1226:
1215:
1213:
1211:
1207:
1202:
1200:
1181:
1165:
1146:
1130:
1126:
1107:
1092:
1087:
1063:
1059:
1055:
1009:
1005:
969:
953:
949:
940:
902:
894:
878:
870:
852:
821:
817:
794:
790:
786:
783:
780:
775:
772:
769:
765:
759:
756:
753:
749:
745:
740:
736:
727:
724:
720:
716:
700:
693:contained in
692:
688:
672:
665:
661:
657:
645:
640:
638:
633:
631:
626:
625:
623:
622:
617:
616:
610:
606:
605:
604:
603:
602:
597:
596:
595:
590:
589:
588:
581:
577:
575:
571:
569:
565:
563:
562:Division ring
559:
558:
557:
556:
550:
545:
544:
516:
500:
498:
492:
476:
462:
461:-adic numbers
460:
455:
439:
425:
423:
418:
416:
412:
410:
403:
401:
397:
396:
395:
394:
393:
384:
380:
378:
374:
370:
366:
365:
364:
360:
358:
354:
352:
348:
346:
342:
340:
336:
334:
330:
329:
328:
324:
323:
322:
321:
315:
310:
309:
300:
296:
295:
294:
290:
286:
282:
280:
276:
275:
274:
270:
266:
262:
261:
260:
256:
255:
254:
253:
228:
224:
215:
212:
205:
204:Terminal ring
201:
178:
174:
173:
172:
168:
166:
162:
160:
156:
154:
150:
148:
144:
143:
142:
141:
140:
133:
129:
127:
123:
121:
117:
116:
115:
114:
113:
106:
102:
100:
96:
94:
90:
86:
82:
80:
76:
75:
74:
73:Quotient ring
70:
68:
64:
62:
58:
57:
56:
55:
46:
45:
42:
37:→ Ring theory
36:
31:
19:
2761:
2713:
2679:Local fields
2678:
2650:
2636:
2628:Cassels 1986
2623:
2611:
2599:
2580:
2574:
2562:
2550:
2520:
2498:Cassels 1986
2478:
2466:
2421:
2417:
2382:
2378:
2365:
2361:
2352:
2345:
2335:
2329:
2323:
2311:
2306:
2283:
2279:
2272:
2268:prime ideals
2257:
2166:
2155:
2051:
2042:
2027:
2018:
1906:
1894:
1890:
1886:
1883:is given by
1879:
1874:
1861:
1857:
1853:
1839:
1828:
1740:
1670:
1550:
1544:
1475:
1469:
1463:
1459:discriminant
1453:
1449:
1446:
1430:
1416:
1409:
1393:
1388:
1384:
1381:
1304:
1290:vector space
1284:
1277:
1271:
1267:
1260:
1255:, that is a
1252:
1247:
1234:
1224:
1219:
1203:
1088:
970:
937:is always a
867:. Since any
726:coefficients
659:
653:
613:
599:
598:
594:Free algebra
592:
591:
585:
584:
553:
496:
458:
421:
414:
390:
389:
369:Finite field
318:
265:Finite field
251:
250:
177:Initial ring
137:
136:
110:
109:
52:
2780:Ring theory
2616:Samuel 1972
2604:Samuel 1972
2567:Samuel 1972
2555:Samuel 1972
2513:Samuel 1972
2023:) /2)
1843: is a
1676:spanned by
915:, the ring
871:belongs to
656:mathematics
574:Simple ring
285:Jordan ring
159:Graded ring
41:Ring theory
18:Number ring
2774:Categories
2750:0956.11021
2702:0595.12006
2641:References
1414:as a free
1216:Properties
891:and is an
580:Commutator
339:GCD domain
2458:Citations
2341:≤ 1
2232:−
2227:−
2210:−
2190:⋅
2131:∈
2084:∈
2014:(1, (1 +
1772:α
1765:…
1753:α
1712:α
1705:…
1685:α
1665:submodule
1611:α
1604:…
1592:α
1569:Δ
1516:∈
1507:α
1500:…
1488:α
1327:∑
784:⋯
773:−
757:−
521:∞
299:Semifield
2760:(1972).
2712:(1999).
2676:(1986).
2391:See also
1864: )
1632:. Then,
1438:Examples
1424:of
293:Semiring
279:Lie ring
61:Subrings
2742:1697859
2581:Algebra
2537:Baker.
2448:integer
2296:. The
2040:√
2036:and by
2016:√
1977:, then
1893:, ...,
1674:-module
1667:of the
1457:is the
1397:. The
1282:of the
1266:, ...,
1052:is the
939:subring
869:integer
723:integer
689:of all
685:is the
495:Prüfer
97:•
2748:
2740:
2730:
2700:
2690:
2662:
2587:
2264:ideals
2114:where
1554:, set
1479:, and
1422:degree
1401:
1292:
1240:module
1127:whose
1030:where
662:of an
658:, the
147:Module
120:Kernel
2542:(PDF)
2409:Notes
2288:is a
2275:units
2266:into
2030:≡ 1 (
1927:is a
1835:prime
1833:is a
1663:is a
1548:over
1473:over
1461:. If
1428:over
1382:with
1257:basis
1229:is a
1206:order
1197:is a
1054:field
721:with
717:of a
499:-ring
363:Field
259:Field
67:Ideal
54:Rings
2728:ISBN
2688:ISBN
2660:ISBN
2585:ISBN
2273:The
2038:(1,
1931:and
1885:(1,
1851:and
1399:rank
1244:free
715:root
687:ring
2746:Zbl
2698:Zbl
2304:of
2292:by
2049:if
2032:mod
2025:if
1907:If
1847:th
1829:If
1821:.
1405:of
1300:in
1275:∈ O
1201:.
1162:of
1056:of
941:of
895:of
836:or
654:In
2776::
2744:.
2738:MR
2736:.
2726:.
2716:.
2696:.
2686:.
2658:.
2654:.
2529:^
2505:^
2490:^
2387:.
2316:.
2270:.
2148:.
2034:4)
1899:.
1889:,
1878:=
1856:=
1837:,
1434:.
1392:∈
1212:.
968:.
728::
607:•
578:•
572:•
566:•
560:•
493:•
456:•
419:•
413:•
404:•
398:•
381:•
375:•
367:•
361:•
355:•
349:•
343:•
337:•
331:•
325:•
297:•
291:•
283:•
277:•
271:•
263:•
257:•
202:•
175:•
169:•
163:•
157:•
151:•
145:•
130:•
124:•
118:•
103:•
91:•
83:•
77:•
71:•
65:•
59:•
2752:.
2704:.
2668:.
2593:.
2433:Z
2383:p
2379:Q
2373:p
2366:p
2362:Z
2356:p
2346:F
2336:F
2330:F
2307:K
2284:K
2280:O
2243:.
2240:)
2235:5
2224:1
2221:(
2218:)
2213:5
2205:+
2202:1
2199:(
2196:=
2193:3
2187:2
2184:=
2181:6
2167:Z
2135:Q
2128:b
2125:,
2122:a
2102:)
2097:d
2092:(
2088:Q
2079:d
2074:b
2071:+
2068:a
2052:d
2047:)
2043:d
2028:d
2019:d
1994:K
1988:O
1961:)
1955:d
1950:(
1946:Q
1942:=
1939:K
1915:d
1897:)
1895:ζ
1891:ζ
1887:ζ
1880:Z
1875:K
1872:O
1862:ζ
1860:(
1858:Q
1854:K
1845:p
1840:ζ
1831:p
1807:K
1801:O
1776:n
1768:,
1762:,
1757:1
1741:d
1726:d
1722:/
1716:n
1708:,
1702:,
1699:d
1695:/
1689:1
1671:Z
1649:K
1643:O
1620:)
1615:n
1607:,
1601:,
1596:1
1588:(
1582:Q
1577:/
1573:K
1565:=
1562:d
1551:Q
1545:K
1528:K
1522:O
1511:n
1503:,
1497:,
1492:1
1476:Q
1470:n
1464:K
1454:Q
1452:/
1450:K
1431:Q
1426:K
1417:Z
1410:K
1407:O
1403:n
1394:Z
1389:i
1385:a
1367:,
1362:i
1358:b
1352:i
1348:a
1342:n
1337:1
1334:=
1331:i
1323:=
1320:x
1305:K
1302:O
1298:x
1294:K
1288:-
1285:Q
1278:K
1272:n
1268:b
1264:1
1261:b
1248:Z
1238:-
1235:Z
1225:K
1222:O
1185:]
1182:i
1179:[
1175:Z
1150:)
1147:i
1144:(
1140:Q
1111:]
1108:i
1105:[
1101:Z
1073:Z
1039:Q
1015:Q
1010:O
1006:=
1002:Z
980:Z
954:K
950:O
924:Z
903:K
879:K
853:K
847:O
822:K
818:O
795:0
791:c
787:+
781:+
776:1
770:n
766:x
760:1
754:n
750:c
746:+
741:n
737:x
701:K
673:K
643:e
636:t
629:v
526:)
517:p
513:(
509:Z
497:p
477:p
472:Q
459:p
440:p
435:Z
422:p
408:n
233:Z
229:1
225:/
220:Z
216:=
213:0
187:Z
20:)
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