2338:
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.
1619:
1529:
2242:
2101:
1725:
796:
1777:
1960:
1017:
1366:
233:
525:
2135:
1995:
1808:
1650:
854:
478:
441:
1149:
2433:
1184:
1110:
1073:
1039:
980:
924:
187:
955:
823:
1914:
902:
878:
700:
672:
1546:
2638:
630:
1471:
1447:
2165:
2052:
2720:
2577:
87:
1668:
720:
2435:
of "ordinary" integers, the prototypical object for all those rings. It is a consequence of the ambiguity of the word "
2680:
2652:
2385:
2046:
1736:
623:
575:
2773:
2712:
2301:
2278:
1923:
985:
1304:
197:
1155:, consisting of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers,
492:
2666:
2150:
616:
483:
2314:
2282:
2106:
1410:
333:
1969:
1782:
1624:
828:
2672:
2644:
93:
454:
417:
108:
1219:
1123:
1117:
568:
371:
321:
2416:
1158:
1084:
1056:
1022:
963:
907:
170:
1050:
380:
114:
73:
1436:
A useful tool for computing the integral closure of the ring of integers in an algebraic field
1245:
652:
537:
388:
339:
120:
1387:
2768:
2730:
1228:
933:
801:
703:
261:
135:
2738:
2690:
8:
2252:
2146:
1917:
1194:
1042:
543:
351:
302:
247:
141:
127:
55:
23:
2527:
2263:
2020:
1899:
887:
863:
685:
675:
657:
556:
42:
2698:
2716:
2676:
2648:
2573:
1998:
1152:
679:
597:
394:
159:
100:
2734:
2686:
2662:
2390:
2286:
1856:
1187:
1079:
881:
707:
603:
589:
345:
308:
81:
67:
2726:
2248:
1963:
1198:
1046:
365:
315:
153:
2159:, the element 6 has two essentially different factorizations into irreducibles:
2343:
2290:
1113:
409:
2762:
2746:
2360:
1837:
550:
446:
61:
1823:
1614:{\displaystyle d=\Delta _{K/\mathbb {Q} }(\alpha _{1},\ldots ,\alpha _{n})}
1278:
582:
357:
253:
2436:
2256:
1232:
714:
644:
562:
273:
147:
29:
2453:
1193:
The ring of integers of an algebraic number field is the unique maximal
2465:
327:
2671:. London Mathematical Society Student Texts. Vol. 3. Cambridge:
1653:
1524:{\displaystyle \alpha _{1},\ldots ,\alpha _{n}\in {\mathcal {O}}_{K}}
287:
192:
281:
267:
927:
857:
711:
165:
49:
2332:; this is a ring because of the strong triangle inequality. If
2145:
In a ring of integers, every element has a factorization into
1120:
are integers. It is the ring of integers in the number field
2237:{\displaystyle 6=2\cdot 3=(1+{\sqrt {-5}})(1-{\sqrt {-5}}).}
2096:{\displaystyle a+b{\sqrt {d}}\in \mathbf {Q} ({\sqrt {d}})}
1075:
are often called the "rational integers" because of this.
2482:
2480:
2598:
2299:. A set of torsion-free generators is called a set of
2497:
2495:
2610:
2477:
2419:
2168:
2109:
2055:
1972:
1926:
1902:
1785:
1739:
1671:
1627:
1549:
1474:
1307:
1161:
1126:
1087:
1059:
1025:
988:
966:
936:
910:
890:
866:
831:
804:
723:
688:
660:
495:
457:
420:
200:
173:
2586:
2549:
2537:
2492:
2413:, without specifying the field, refers to the ring
2393:– gives a technique for computing integral closures
1720:{\displaystyle \alpha _{1}/d,\ldots ,\alpha _{n}/d}
982:is the simplest possible ring of integers. Namely,
2427:
2236:
2129:
2095:
1989:
1954:
1908:
1802:
1771:
1719:
1644:
1613:
1523:
1360:
1178:
1143:
1104:
1067:
1033:
1011:
974:
949:
918:
896:
872:
848:
817:
791:{\displaystyle x^{n}+c_{n-1}x^{n-1}+\cdots +c_{0}}
790:
694:
666:
519:
472:
435:
227:
181:
2760:
1772:{\displaystyle \alpha _{1},\ldots ,\alpha _{n}}
2708:Grundlehren der mathematischen Wissenschaften
2636:
2471:
2459:
2149:, but the ring need not have the property of
1955:{\displaystyle K=\mathbb {Q} ({\sqrt {d}}\,)}
1012:{\displaystyle \mathbb {Z} =O_{\mathbb {Q} }}
624:
2706:
1361:{\displaystyle x=\sum _{i=1}^{n}a_{i}b_{i},}
228:{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }
2637:Alaca, Saban; Williams, Kenneth S. (2003).
2140:
631:
617:
2421:
1948:
1934:
1570:
1163:
1128:
1089:
1078:The next simplest example is the ring of
1061:
1027:
1003:
990:
968:
912:
520:{\displaystyle \mathbb {Z} (p^{\infty })}
497:
460:
423:
221:
208:
175:
2697:
1813:
2661:
2616:
2486:
2153:: for example, in the ring of integers
1891:
2761:
2745:
2604:
2592:
2555:
2543:
2501:
2313:One defines the ring of integers of a
2567:
2507:
2251:, and so has unique factorization of
2045:. This can be found by computing the
1431:
2640:Introductory Algebraic Number Theory
2521:
2519:
88:Free product of associative algebras
2130:{\displaystyle a,b\in \mathbf {Q} }
2001:and its integral basis is given by
13:
1990:{\displaystyle {\mathcal {O}}_{K}}
1976:
1803:{\displaystyle {\mathcal {O}}_{K}}
1789:
1645:{\displaystyle {\mathcal {O}}_{K}}
1631:
1557:
1510:
849:{\displaystyle {\mathcal {O}}_{K}}
835:
509:
14:
2785:
2525:
2516:
2386:Minimal polynomial (field theory)
2308:
576:Noncommutative algebraic geometry
2359:are the ring of integers of the
2279:finitely generated abelian group
2123:
2076:
798:. This ring is often denoted by
473:{\displaystyle \mathbb {Q} _{p}}
436:{\displaystyle \mathbb {Z} _{p}}
2561:
2247:A ring of integers is always a
1298:can be uniquely represented as
1144:{\displaystyle \mathbb {Q} (i)}
2572:. Prentice Hall. p. 360.
2404:
2322:as the set of all elements of
2228:
2209:
2206:
2187:
2090:
2080:
1949:
1938:
1608:
1576:
1173:
1167:
1138:
1132:
1099:
1093:
514:
501:
1:
2629:
2462:, p. 110, Defs. 6.1.2-3.
1204:
1197:in the field. It is always a
2474:, p. 74, Defs. 4.1.1-2.
2446:
2428:{\displaystyle \mathbb {Z} }
1859:, then an integral basis of
1779:forms an integral basis for
1285:such that each element
1179:{\displaystyle \mathbb {Z} }
1105:{\displaystyle \mathbb {Z} }
1068:{\displaystyle \mathbb {Z} }
1034:{\displaystyle \mathbb {Q} }
975:{\displaystyle \mathbb {Z} }
919:{\displaystyle \mathbb {Z} }
702:. An algebraic integer is a
182:{\displaystyle \mathbb {Z} }
7:
2379:
2315:non-archimedean local field
1426:
334:Unique factorization domain
10:
2790:
2703:Algebraische Zahlentheorie
2673:Cambridge University Press
2645:Cambridge University Press
94:Tensor product of algebras
2711:. Vol. 322. Berlin:
2528:"Algebraic Number Theory"
2472:Alaca & Williams 2003
2460:Alaca & Williams 2003
1240:-module, and thus has an
2397:
2283:Dirichlet's unit theorem
2141:Multiplicative structure
2049:of an arbitrary element
1409:-module is equal to the
1118:real and imaginary parts
372:Formal power series ring
322:Integrally closed domain
2774:Algebraic number theory
2751:Algebraic number theory
2568:Artin, Michael (2011).
1051:algebraic number theory
381:Algebraic number theory
74:Total ring of fractions
2707:
2439:" in abstract algebra.
2429:
2266:of a ring of integers
2238:
2131:
2097:
1991:
1956:
1910:
1804:
1773:
1721:
1646:
1615:
1525:
1362:
1334:
1180:
1145:
1106:
1069:
1035:
1013:
976:
951:
920:
898:
874:
850:
819:
792:
696:
668:
653:algebraic number field
538:Noncommutative algebra
521:
474:
437:
389:Algebraic number field
340:Principal ideal domain
229:
183:
121:Frobenius endomorphism
16:Algebraic construction
2513:Cassels (1986) p. 193
2430:
2239:
2132:
2098:
1992:
1962:is the corresponding
1957:
1911:
1855:is the corresponding
1814:Cyclotomic extensions
1805:
1774:
1733:is square-free, then
1722:
1647:
1616:
1526:
1363:
1314:
1209:The ring of integers
1181:
1146:
1107:
1070:
1036:
1014:
977:
960:The ring of integers
952:
950:{\displaystyle O_{K}}
921:
899:
875:
851:
820:
818:{\displaystyle O_{K}}
793:
697:
669:
522:
475:
438:
230:
184:
2417:
2411:The ring of integers
2328:with absolute value
2166:
2151:unique factorization
2147:irreducible elements
2107:
2053:
1970:
1924:
1900:
1892:Quadratic extensions
1783:
1737:
1669:
1625:
1547:
1472:
1305:
1159:
1124:
1085:
1057:
1023:
986:
964:
934:
908:
888:
864:
829:
802:
721:
686:
658:
544:Noncommutative rings
493:
455:
418:
262:Non-associative ring
198:
171:
128:Algebraic structures
1918:square-free integer
1231:. Indeed, it is a
303:Commutative algebra
142:Associative algebra
24:Algebraic structure
2753:. Hermann/Kershaw.
2425:
2234:
2127:
2093:
2047:minimal polynomial
1999:quadratic integers
1987:
1952:
1906:
1800:
1769:
1717:
1642:
1611:
1521:
1432:Computational tool
1358:
1220:finitely-generated
1176:
1153:Gaussian rationals
1141:
1102:
1065:
1049:. And indeed, in
1031:
1009:
972:
947:
916:
894:
870:
846:
815:
788:
692:
680:algebraic integers
664:
557:Semiprimitive ring
517:
470:
433:
241:Related structures
225:
179:
115:Inner automorphism
101:Ring homomorphisms
2722:978-3-540-65399-8
2607:, pp. 59–62.
2579:978-0-13-241377-0
2533:. pp. 33–35.
2342:For example, the
2302:fundamental units
2226:
2204:
2088:
2070:
1946:
1909:{\displaystyle d}
1080:Gaussian integers
897:{\displaystyle K}
873:{\displaystyle K}
695:{\displaystyle K}
667:{\displaystyle K}
641:
640:
598:Geometric algebra
309:Commutative rings
160:Category of rings
2781:
2754:
2742:
2710:
2699:Neukirch, Jürgen
2694:
2658:
2620:
2614:
2608:
2602:
2596:
2590:
2584:
2583:
2565:
2559:
2553:
2547:
2541:
2535:
2534:
2532:
2523:
2514:
2511:
2505:
2499:
2490:
2484:
2475:
2469:
2463:
2457:
2440:
2434:
2432:
2431:
2426:
2424:
2408:
2391:Integral closure
2375:
2363:
2358:
2346:
2337:
2331:
2327:
2321:
2298:
2289:consists of the
2287:torsion subgroup
2276:
2243:
2241:
2240:
2235:
2227:
2219:
2205:
2197:
2158:
2136:
2134:
2133:
2128:
2126:
2102:
2100:
2099:
2094:
2089:
2084:
2079:
2071:
2066:
2044:
2037:
2035:
2034:
2024:
2013:
2011:
2010:
1996:
1994:
1993:
1988:
1986:
1985:
1980:
1979:
1961:
1959:
1958:
1953:
1947:
1942:
1937:
1915:
1913:
1912:
1907:
1887:
1871:
1857:cyclotomic field
1854:
1835:
1831:
1821:
1809:
1807:
1806:
1801:
1799:
1798:
1793:
1792:
1778:
1776:
1775:
1770:
1768:
1767:
1749:
1748:
1732:
1726:
1724:
1723:
1718:
1713:
1708:
1707:
1686:
1681:
1680:
1664:
1662:
1651:
1649:
1648:
1643:
1641:
1640:
1635:
1634:
1620:
1618:
1617:
1612:
1607:
1606:
1588:
1587:
1575:
1574:
1573:
1568:
1542:
1536:
1531:form a basis of
1530:
1528:
1527:
1522:
1520:
1519:
1514:
1513:
1503:
1502:
1484:
1483:
1467:
1461:
1455:
1445:
1422:
1416:
1408:
1402:
1393:
1385:
1367:
1365:
1364:
1359:
1354:
1353:
1344:
1343:
1333:
1328:
1297:
1288:
1284:
1276:
1270:
1239:
1226:
1217:
1188:Euclidean domain
1185:
1183:
1182:
1177:
1166:
1150:
1148:
1147:
1142:
1131:
1112:, consisting of
1111:
1109:
1108:
1103:
1092:
1074:
1072:
1071:
1066:
1064:
1053:the elements of
1047:rational numbers
1040:
1038:
1037:
1032:
1030:
1018:
1016:
1015:
1010:
1008:
1007:
1006:
993:
981:
979:
978:
973:
971:
956:
954:
953:
948:
946:
945:
925:
923:
922:
917:
915:
903:
901:
900:
895:
882:integral element
879:
877:
876:
871:
855:
853:
852:
847:
845:
844:
839:
838:
824:
822:
821:
816:
814:
813:
797:
795:
794:
789:
787:
786:
768:
767:
752:
751:
733:
732:
708:monic polynomial
701:
699:
698:
693:
673:
671:
670:
665:
649:ring of integers
633:
626:
619:
604:Operator algebra
590:Clifford algebra
526:
524:
523:
518:
513:
512:
500:
479:
477:
476:
471:
469:
468:
463:
442:
440:
439:
434:
432:
431:
426:
404:Ring of integers
398:
395:Integers modulo
346:Euclidean domain
234:
232:
231:
226:
224:
216:
211:
188:
186:
185:
180:
178:
82:Product of rings
68:Fractional ideal
27:
19:
18:
2789:
2788:
2784:
2783:
2782:
2780:
2779:
2778:
2759:
2758:
2757:
2723:
2713:Springer-Verlag
2683:
2663:Cassels, J.W.S.
2655:
2632:
2626:
2624:
2623:
2615:
2611:
2603:
2599:
2591:
2587:
2580:
2566:
2562:
2554:
2550:
2542:
2538:
2530:
2524:
2517:
2512:
2508:
2500:
2493:
2485:
2478:
2470:
2466:
2458:
2454:
2449:
2444:
2443:
2420:
2418:
2415:
2414:
2409:
2405:
2400:
2382:
2374:
2366:
2361:
2357:
2349:
2344:
2333:
2329:
2323:
2317:
2311:
2294:
2275:
2267:
2249:Dedekind domain
2218:
2196:
2167:
2164:
2163:
2154:
2143:
2122:
2108:
2105:
2104:
2083:
2075:
2065:
2054:
2051:
2050:
2039:
2030:
2028:
2026:
2015:
2006:
2004:
2002:
1981:
1975:
1974:
1973:
1971:
1968:
1967:
1964:quadratic field
1941:
1933:
1925:
1922:
1921:
1901:
1898:
1897:
1894:
1873:
1866:
1860:
1841:
1833:
1827:
1819:
1816:
1794:
1788:
1787:
1786:
1784:
1781:
1780:
1763:
1759:
1744:
1740:
1738:
1735:
1734:
1728:
1727:. In fact, if
1709:
1703:
1699:
1682:
1676:
1672:
1670:
1667:
1666:
1658:
1657:
1636:
1630:
1629:
1628:
1626:
1623:
1622:
1602:
1598:
1583:
1579:
1569:
1564:
1560:
1556:
1548:
1545:
1544:
1538:
1532:
1515:
1509:
1508:
1507:
1498:
1494:
1479:
1475:
1473:
1470:
1469:
1463:
1457:
1451:
1437:
1434:
1429:
1418:
1414:
1404:
1401:
1395:
1391:
1380:
1372:
1349:
1345:
1339:
1335:
1329:
1318:
1306:
1303:
1302:
1296:
1290:
1286:
1282:
1272:
1269:
1263:
1254:
1248:
1235:
1222:
1216:
1210:
1207:
1199:Dedekind domain
1162:
1160:
1157:
1156:
1127:
1125:
1122:
1121:
1114:complex numbers
1088:
1086:
1083:
1082:
1060:
1058:
1055:
1054:
1026:
1024:
1021:
1020:
1002:
1001:
997:
989:
987:
984:
983:
967:
965:
962:
961:
941:
937:
935:
932:
931:
911:
909:
906:
905:
889:
886:
885:
865:
862:
861:
840:
834:
833:
832:
830:
827:
826:
809:
805:
803:
800:
799:
782:
778:
757:
753:
741:
737:
728:
724:
722:
719:
718:
687:
684:
683:
659:
656:
655:
637:
608:
607:
540:
530:
529:
508:
504:
496:
494:
491:
490:
464:
459:
458:
456:
453:
452:
427:
422:
421:
419:
416:
415:
396:
366:Polynomial ring
316:Integral domain
305:
295:
294:
220:
212:
207:
199:
196:
195:
174:
172:
169:
168:
154:Involutive ring
39:
28:
22:
17:
12:
11:
5:
2787:
2777:
2776:
2771:
2756:
2755:
2747:Samuel, Pierre
2743:
2721:
2695:
2681:
2659:
2653:
2633:
2631:
2628:
2622:
2621:
2609:
2597:
2585:
2578:
2560:
2548:
2536:
2515:
2506:
2491:
2489:, p. 192.
2476:
2464:
2451:
2450:
2448:
2445:
2442:
2441:
2423:
2402:
2401:
2399:
2396:
2395:
2394:
2388:
2381:
2378:
2370:
2353:
2347:-adic integers
2310:
2309:Generalization
2307:
2291:roots of unity
2271:
2245:
2244:
2233:
2230:
2225:
2222:
2217:
2214:
2211:
2208:
2203:
2200:
2195:
2192:
2189:
2186:
2183:
2180:
2177:
2174:
2171:
2142:
2139:
2125:
2121:
2118:
2115:
2112:
2092:
2087:
2082:
2078:
2074:
2069:
2064:
2061:
2058:
2043:≡ 2, 3 (mod 4)
1984:
1978:
1951:
1945:
1940:
1936:
1932:
1929:
1905:
1893:
1890:
1862:
1815:
1812:
1797:
1791:
1766:
1762:
1758:
1755:
1752:
1747:
1743:
1716:
1712:
1706:
1702:
1698:
1695:
1692:
1689:
1685:
1679:
1675:
1639:
1633:
1610:
1605:
1601:
1597:
1594:
1591:
1586:
1582:
1578:
1572:
1567:
1563:
1559:
1555:
1552:
1518:
1512:
1506:
1501:
1497:
1493:
1490:
1487:
1482:
1478:
1433:
1430:
1428:
1425:
1397:
1376:
1369:
1368:
1357:
1352:
1348:
1342:
1338:
1332:
1327:
1324:
1321:
1317:
1313:
1310:
1292:
1265:
1259:
1252:
1242:integral basis
1212:
1206:
1203:
1175:
1172:
1169:
1165:
1140:
1137:
1134:
1130:
1101:
1098:
1095:
1091:
1063:
1029:
1005:
1000:
996:
992:
970:
944:
940:
914:
893:
869:
843:
837:
812:
808:
785:
781:
777:
774:
771:
766:
763:
760:
756:
750:
747:
744:
740:
736:
731:
727:
691:
663:
639:
638:
636:
635:
628:
621:
613:
610:
609:
601:
600:
572:
571:
565:
559:
553:
541:
536:
535:
532:
531:
528:
527:
516:
511:
507:
503:
499:
480:
467:
462:
443:
430:
425:
413:-adic integers
406:
400:
391:
377:
376:
375:
374:
368:
362:
361:
360:
348:
342:
336:
330:
324:
306:
301:
300:
297:
296:
293:
292:
291:
290:
278:
277:
276:
270:
258:
257:
256:
238:
237:
236:
235:
223:
219:
215:
210:
206:
203:
189:
177:
156:
150:
144:
138:
124:
123:
117:
111:
97:
96:
90:
84:
78:
77:
76:
70:
58:
52:
40:
38:Basic concepts
37:
36:
33:
32:
15:
9:
6:
4:
3:
2:
2786:
2775:
2772:
2770:
2767:
2766:
2764:
2752:
2748:
2744:
2740:
2736:
2732:
2728:
2724:
2718:
2714:
2709:
2704:
2700:
2696:
2692:
2688:
2684:
2682:0-521-31525-5
2678:
2674:
2670:
2669:
2664:
2660:
2656:
2654:9780511791260
2650:
2646:
2642:
2641:
2635:
2634:
2627:
2619:, p. 41.
2618:
2613:
2606:
2601:
2595:, p. 50.
2594:
2589:
2581:
2575:
2571:
2564:
2558:, p. 35.
2557:
2552:
2546:, p. 43.
2545:
2540:
2529:
2522:
2520:
2510:
2504:, p. 49.
2503:
2498:
2496:
2488:
2483:
2481:
2473:
2468:
2461:
2456:
2452:
2438:
2412:
2407:
2403:
2392:
2389:
2387:
2384:
2383:
2377:
2373:
2369:
2365:
2364:-adic numbers
2356:
2352:
2348:
2340:
2336:
2326:
2320:
2316:
2306:
2304:
2303:
2297:
2292:
2288:
2284:
2280:
2274:
2270:
2265:
2260:
2258:
2254:
2250:
2231:
2223:
2220:
2215:
2212:
2201:
2198:
2193:
2190:
2184:
2181:
2178:
2175:
2172:
2169:
2162:
2161:
2160:
2157:
2152:
2148:
2138:
2119:
2116:
2113:
2110:
2085:
2072:
2067:
2062:
2059:
2056:
2048:
2042:
2033:
2022:
2018:
2009:
2000:
1997:is a ring of
1982:
1965:
1943:
1930:
1927:
1919:
1903:
1889:
1885:
1881:
1877:
1870:
1865:
1858:
1852:
1848:
1844:
1839:
1838:root of unity
1830:
1825:
1811:
1795:
1764:
1760:
1756:
1753:
1750:
1745:
1741:
1731:
1714:
1710:
1704:
1700:
1696:
1693:
1690:
1687:
1683:
1677:
1673:
1661:
1655:
1637:
1603:
1599:
1595:
1592:
1589:
1584:
1580:
1565:
1561:
1553:
1550:
1541:
1535:
1516:
1504:
1499:
1495:
1491:
1488:
1485:
1480:
1476:
1466:
1460:
1456:is of degree
1454:
1449:
1444:
1440:
1424:
1421:
1412:
1407:
1400:
1389:
1384:
1379:
1375:
1355:
1350:
1346:
1340:
1336:
1330:
1325:
1322:
1319:
1315:
1311:
1308:
1301:
1300:
1299:
1295:
1280:
1275:
1268:
1262:
1258:
1251:
1247:
1243:
1238:
1234:
1230:
1225:
1221:
1215:
1202:
1200:
1196:
1191:
1189:
1170:
1154:
1135:
1119:
1115:
1096:
1081:
1076:
1052:
1048:
1044:
998:
994:
958:
942:
938:
929:
891:
883:
867:
859:
841:
810:
806:
783:
779:
775:
772:
769:
764:
761:
758:
754:
748:
745:
742:
738:
734:
729:
725:
716:
713:
709:
705:
689:
682:contained in
681:
677:
661:
654:
650:
646:
634:
629:
627:
622:
620:
615:
614:
612:
611:
606:
605:
599:
595:
594:
593:
592:
591:
586:
585:
584:
579:
578:
577:
570:
566:
564:
560:
558:
554:
552:
551:Division ring
548:
547:
546:
545:
539:
534:
533:
505:
489:
487:
481:
465:
451:
450:-adic numbers
449:
444:
428:
414:
412:
407:
405:
401:
399:
392:
390:
386:
385:
384:
383:
382:
373:
369:
367:
363:
359:
355:
354:
353:
349:
347:
343:
341:
337:
335:
331:
329:
325:
323:
319:
318:
317:
313:
312:
311:
310:
304:
299:
298:
289:
285:
284:
283:
279:
275:
271:
269:
265:
264:
263:
259:
255:
251:
250:
249:
245:
244:
243:
242:
217:
213:
204:
201:
194:
193:Terminal ring
190:
167:
163:
162:
161:
157:
155:
151:
149:
145:
143:
139:
137:
133:
132:
131:
130:
129:
122:
118:
116:
112:
110:
106:
105:
104:
103:
102:
95:
91:
89:
85:
83:
79:
75:
71:
69:
65:
64:
63:
62:Quotient ring
59:
57:
53:
51:
47:
46:
45:
44:
35:
34:
31:
26:→ Ring theory
25:
21:
20:
2750:
2702:
2668:Local fields
2667:
2639:
2625:
2617:Cassels 1986
2612:
2600:
2588:
2569:
2563:
2551:
2539:
2509:
2487:Cassels 1986
2467:
2455:
2410:
2406:
2371:
2367:
2354:
2350:
2341:
2334:
2324:
2318:
2312:
2300:
2295:
2272:
2268:
2261:
2257:prime ideals
2246:
2155:
2144:
2040:
2031:
2016:
2007:
1895:
1883:
1879:
1875:
1872:is given by
1868:
1863:
1850:
1846:
1842:
1828:
1817:
1729:
1659:
1539:
1533:
1464:
1458:
1452:
1448:discriminant
1442:
1438:
1435:
1419:
1405:
1398:
1382:
1377:
1373:
1370:
1293:
1279:vector space
1273:
1266:
1260:
1256:
1249:
1244:, that is a
1241:
1236:
1223:
1213:
1208:
1192:
1077:
959:
926:is always a
856:. Since any
715:coefficients
648:
642:
602:
588:
587:
583:Free algebra
581:
580:
574:
573:
542:
485:
447:
410:
403:
379:
378:
358:Finite field
307:
254:Finite field
240:
239:
166:Initial ring
126:
125:
99:
98:
41:
2769:Ring theory
2605:Samuel 1972
2593:Samuel 1972
2556:Samuel 1972
2544:Samuel 1972
2502:Samuel 1972
2012:) /2)
1832: is a
1665:spanned by
904:, the ring
860:belongs to
645:mathematics
563:Simple ring
274:Jordan ring
148:Graded ring
30:Ring theory
2763:Categories
2739:0956.11021
2691:0595.12006
2630:References
1403:as a free
1205:Properties
880:and is an
569:Commutator
328:GCD domain
2447:Citations
2330:≤ 1
2221:−
2216:−
2199:−
2179:⋅
2120:∈
2073:∈
2003:(1, (1 +
1761:α
1754:…
1742:α
1701:α
1694:…
1674:α
1654:submodule
1600:α
1593:…
1581:α
1558:Δ
1505:∈
1496:α
1489:…
1477:α
1316:∑
773:⋯
762:−
746:−
510:∞
288:Semifield
2749:(1972).
2701:(1999).
2665:(1986).
2380:See also
1853: )
1621:. Then,
1427:Examples
1413:of
282:Semiring
268:Lie ring
50:Subrings
2731:1697859
2570:Algebra
2526:Baker.
2437:integer
2285:. The
2029:√
2025:and by
2005:√
1966:, then
1882:, ...,
1663:-module
1656:of the
1446:is the
1386:. The
1271:of the
1255:, ...,
1041:is the
928:subring
858:integer
712:integer
678:of all
674:is the
484:Prüfer
86:•
2737:
2729:
2719:
2689:
2679:
2651:
2576:
2253:ideals
2103:where
1543:, set
1468:, and
1411:degree
1390:
1281:
1229:module
1116:whose
1019:where
651:of an
647:, the
136:Module
109:Kernel
2531:(PDF)
2398:Notes
2277:is a
2264:units
2255:into
2019:≡ 1 (
1916:is a
1824:prime
1822:is a
1652:is a
1537:over
1462:over
1450:. If
1417:over
1371:with
1246:basis
1218:is a
1195:order
1186:is a
1043:field
710:with
706:of a
488:-ring
352:Field
248:Field
56:Ideal
43:Rings
2717:ISBN
2677:ISBN
2649:ISBN
2574:ISBN
2262:The
2027:(1,
1920:and
1874:(1,
1840:and
1388:rank
1233:free
704:root
676:ring
2735:Zbl
2687:Zbl
2293:of
2281:by
2038:if
2021:mod
2014:if
1896:If
1836:th
1818:If
1810:.
1394:of
1289:in
1264:∈ O
1190:.
1151:of
1045:of
930:of
884:of
825:or
643:In
2765::
2733:.
2727:MR
2725:.
2715:.
2705:.
2685:.
2675:.
2647:.
2643:.
2518:^
2494:^
2479:^
2376:.
2305:.
2259:.
2137:.
2023:4)
1888:.
1878:,
1867:=
1845:=
1826:,
1423:.
1381:∈
1201:.
957:.
717::
596:•
567:•
561:•
555:•
549:•
482:•
445:•
408:•
402:•
393:•
387:•
370:•
364:•
356:•
350:•
344:•
338:•
332:•
326:•
320:•
314:•
286:•
280:•
272:•
266:•
260:•
252:•
246:•
191:•
164:•
158:•
152:•
146:•
140:•
134:•
119:•
113:•
107:•
92:•
80:•
72:•
66:•
60:•
54:•
48:•
2741:.
2693:.
2657:.
2582:.
2422:Z
2372:p
2368:Q
2362:p
2355:p
2351:Z
2345:p
2335:F
2325:F
2319:F
2296:K
2273:K
2269:O
2232:.
2229:)
2224:5
2213:1
2210:(
2207:)
2202:5
2194:+
2191:1
2188:(
2185:=
2182:3
2176:2
2173:=
2170:6
2156:Z
2124:Q
2117:b
2114:,
2111:a
2091:)
2086:d
2081:(
2077:Q
2068:d
2063:b
2060:+
2057:a
2041:d
2036:)
2032:d
2017:d
2008:d
1983:K
1977:O
1950:)
1944:d
1939:(
1935:Q
1931:=
1928:K
1904:d
1886:)
1884:ζ
1880:ζ
1876:ζ
1869:Z
1864:K
1861:O
1851:ζ
1849:(
1847:Q
1843:K
1834:p
1829:ζ
1820:p
1796:K
1790:O
1765:n
1757:,
1751:,
1746:1
1730:d
1715:d
1711:/
1705:n
1697:,
1691:,
1688:d
1684:/
1678:1
1660:Z
1638:K
1632:O
1609:)
1604:n
1596:,
1590:,
1585:1
1577:(
1571:Q
1566:/
1562:K
1554:=
1551:d
1540:Q
1534:K
1517:K
1511:O
1500:n
1492:,
1486:,
1481:1
1465:Q
1459:n
1453:K
1443:Q
1441:/
1439:K
1420:Q
1415:K
1406:Z
1399:K
1396:O
1392:n
1383:Z
1378:i
1374:a
1356:,
1351:i
1347:b
1341:i
1337:a
1331:n
1326:1
1323:=
1320:i
1312:=
1309:x
1294:K
1291:O
1287:x
1283:K
1277:-
1274:Q
1267:K
1261:n
1257:b
1253:1
1250:b
1237:Z
1227:-
1224:Z
1214:K
1211:O
1174:]
1171:i
1168:[
1164:Z
1139:)
1136:i
1133:(
1129:Q
1100:]
1097:i
1094:[
1090:Z
1062:Z
1028:Q
1004:Q
999:O
995:=
991:Z
969:Z
943:K
939:O
913:Z
892:K
868:K
842:K
836:O
811:K
807:O
784:0
780:c
776:+
770:+
765:1
759:n
755:x
749:1
743:n
739:c
735:+
730:n
726:x
690:K
662:K
632:e
625:t
618:v
515:)
506:p
502:(
498:Z
486:p
466:p
461:Q
448:p
429:p
424:Z
411:p
397:n
222:Z
218:1
214:/
209:Z
205:=
202:0
176:Z
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