36:
329:
generated by those roots of unity was a major obstacle. Out of Kummer's work for the first time came a study of the obstruction to the factorization. We now recognise this as part of the ideal class group: in fact Kummer had isolated the
384:). The size of the ideal class group can be considered as a measure for the deviation of a ring from being a principal ideal domain; a ring is a principal ideal domain if and only if it has a trivial ideal class group.
1071:
737:. The failure of these groups to be trivial is a measure of the failure of the map to be an isomorphism: that is the failure of ideals to act like ring elements, that is to say, like numbers.
1454:
1273:
1229:
1185:
1138:
950:
861:
1791:
1716:
1926:
1677:
625:
less than the bound. In general the bound is not sharp enough to make the calculation practical for fields with large discriminant, but computers are well suited to the task.
2096:
1513:
1614:
1965:
1878:
1548:
894:
364:
formulated the concept of an ideal, Kummer having worked in a different way. At this point the existing examples could be unified. It was shown that while rings of
2040:
1843:
2011:
1985:
1756:
1736:
1638:
1588:
1568:
1376:
970:
914:
821:
1140:
with class number 1. Computational results indicate that there are a great many such fields. However, it is not even known if there are infinitely many
255:, for which the multiplicative properties are intimately tied to the structure of the class group. For example, the class group of a Dedekind domain is
543:. The group property of existence of inverse elements follows easily from the fact that, in a Dedekind domain, every non-zero ideal (except
17:
2265:
614:
1235:> 0, the ideal class group may be half the size since the class group of integral binary quadratic forms is isomorphic to the
100:
686:) is the class group. Higher K groups can also be employed and interpreted arithmetically in connection to rings of integers.
72:
2443:
2404:
79:
975:
274:
Ideal class groups (or, rather, what were effectively ideal class groups) were studied some time before the idea of an
53:
1385:
694:
It was remarked above that the ideal class group provides part of the answer to the question of how much ideals in a
322:
119:
2478:
2435:
1192:
86:
1242:
1198:
1154:
1107:
919:
830:
1761:
1686:
57:
68:
2483:
609:
Computation of the class group is hard, in general; it can be done by hand for the ring of integers in an
1883:
572:
263:
2396:
213:
1643:
2255:
2124:
is not principal is also related to the fact that the element 6 has two distinct factorisations into
1090:
2052:
1459:
702:
of the
Dedekind domain, since passage from principal ideals to their generators requires the use of
2285:
314:
2159:
603:
46:
2270:
1355:
706:(and this is the rest of the reason for introducing the concept of fractional ideal, as well):
610:
568:
524:
401:
369:
145:
93:
1593:
1188:
283:
1935:
1848:
1518:
873:
621:. This result gives a bound, depending on the ring, such that every ideal class contains an
353:, as the reason for the failure of the standard method of attack on the Fermat problem (see
2453:
2414:
2304:
2250:
2245:
2047:
1094:
1078:
618:
465:
287:
217:
2461:
2384:
2016:
1819:
559:
The ideal class group is trivial (i.e. has only one element) if and only if all ideals of
8:
2259:
2175:
2125:
1990:
824:
339:
275:
182:
2299:
2290:
2275:
2155:
2099:
1970:
1741:
1721:
1623:
1573:
1553:
1361:
1236:
955:
899:
806:
753:
718:
703:
637:
326:
310:
252:
2421:
748:
2439:
2400:
2163:
1282:
717:
by sending every element to the principal (fractional) ideal it generates. This is a
591:
531:, the multiplication defined above turns the set of fractional ideal classes into an
520:
365:
291:
2457:
2388:
2371:
2167:
1680:
1290:
777:
759:
722:
485:
405:
361:
335:
306:
227:
190:
186:
476:. Ideal classes can be multiplied: if denotes the equivalence class of the ideal
2449:
2410:
864:
788:
695:
528:
489:
453:
397:
381:
248:
773:
699:
318:
279:
259:
154:
368:
do not always have unique factorization into primes (because they need not be
2472:
2280:
1617:
780:), and so have class number 1: that is, they have trivial ideal class groups.
765:
571:, and hence from satisfying unique prime factorization (Dedekind domains are
532:
354:
309:. It had been realised (probably by several people) that failure to complete
295:
256:
2376:
2295:
2171:
1333:
1141:
1086:
1082:
373:
347:
302:
231:
321:
was for a very good reason: a failure of unique factorization – i.e., the
1279:
548:
481:
377:
133:
794:
is an integral domain. It has a countably infinite set of ideal classes.
2179:
1379:
1074:
622:
2113:
in the ideal class group has order two. Showing that there aren't any
698:
behave like elements. The other part of the answer is provided by the
563:
are principal. In this sense, the ideal class group measures how far
286:
integral quadratic forms, as put into something like a final form by
1104:> 0, then it is unknown whether there are infinitely many fields
35:
730:
209:
633:
598:
is a ring of algebraic integers, then the class number is always
2362:
Claborn, Luther (1966), "Every abelian group is a class group",
484:. The principal ideals form the ideal class which serves as an
509:
590:) may be infinite in general. In fact, every abelian group is
269:
216:
ideals. The class group is a measure of the extent to which
2395:, Cambridge Studies in Advanced Mathematics, vol. 27,
2182:
abelian extension of such a field. The
Hilbert class field
1294:
1286:
636:, and the class group can be subsumed under the heading of
504:
may not exist and consequently the set of ideal classes of
2228:
with Galois group isomorphic to the ideal class group of
1328:
possess unique factorization; in fact the class group of
594:
to the ideal class group of some
Dedekind domain. But if
776:), are all principal ideal domains (and in fact are all
2178:
of a number field, which can be defined as the maximal
1300:
278:
was formulated. These groups appeared in the theory of
2098:
has no solutions in integers, as it has no solutions
2055:
2019:
1993:
1973:
1938:
1886:
1851:
1822:
1764:
1744:
1724:
1689:
1646:
1626:
1596:
1576:
1556:
1521:
1462:
1388:
1364:
1245:
1201:
1157:
1110:
978:
958:
922:
902:
876:
833:
809:
27:
In number theory, measure of non-unique factorization
2150:
952:
is equal to 1 for precisely the following values of
798:
2174:. A particularly beautiful example is found in the
689:
60:. Unsourced material may be challenged and removed.
2298:—a generalisation of the class group appearing in
2090:
2034:
2005:
1979:
1959:
1920:
1872:
1837:
1785:
1750:
1730:
1710:
1671:
1632:
1608:
1582:
1562:
1542:
1507:
1448:
1370:
1267:
1223:
1179:
1132:
1085:, although Heegner's proof was not believed until
1066:{\displaystyle d=-1,-2,-3,-7,-11,-19,-43,-67,-163}
1065:
964:
944:
908:
888:
855:
827:(a product of distinct primes) other than 1, then
815:
575:if and only if they are principal ideal domains).
480:, then the multiplication = is well-defined and
740:
2470:
2236:Neither property is particularly easy to prove.
380:(that is, every ring of algebraic integers is a
1449:{\displaystyle N(a+b{\sqrt {-5}})=a^{2}+5b^{2}}
713:to the set of all nonzero fractional ideals of
602:. This is one of the main results of classical
2383:
2345:
488:for this multiplication. Thus a class has an
376:admits a unique factorization as a product of
2431:Grundlehren der mathematischen Wissenschaften
1187:is isomorphic to the class group of integral
2429:
2190:is unique and has the following properties:
2266:List of number fields with class number one
2166:of a given algebraic number field, meaning
2109:= (2), which is principal, so the class of
1268:{\displaystyle \mathbf {Q} ({\sqrt {d}}\,)}
1224:{\displaystyle \mathbf {Q} ({\sqrt {d}}\,)}
1180:{\displaystyle \mathbf {Q} ({\sqrt {d}}\,)}
1133:{\displaystyle \mathbf {Q} ({\sqrt {d}}\,)}
945:{\displaystyle \mathbf {Q} ({\sqrt {d}}\,)}
856:{\displaystyle \mathbf {Q} ({\sqrt {d}}\,)}
290:, a composition law was defined on certain
270:History and origin of the ideal class group
1786:{\displaystyle \mathbf {Z} /3\mathbf {Z} }
1711:{\displaystyle \mathbf {Z} /6\mathbf {Z} }
2375:
1261:
1217:
1173:
1126:
938:
849:
500:is a principal ideal. In general, such a
468:. The equivalence classes are called the
120:Learn how and when to remove this message
2420:
2334:
1093:.) This is a special case of the famous
372:), they do have the property that every
2361:
2323:
2194:Every ideal of the ring of integers of
2046:has no elements of norm 2, because the
655:its ideal class group; more precisely,
632:to their corresponding class groups is
14:
2471:
464:.) It is easily shown that this is an
420:whenever there exist nonzero elements
2117:ideal classes requires more effort.
1301:Example of a non-trivial class group
1285:rings, the class number is given in
896:, then the class number of the ring
58:adding citations to reliable sources
29:
1921:{\displaystyle N(1+{\sqrt {-5}})=6}
628:The mapping from rings of integers
460:consisting of all the multiples of
24:
25:
2495:
2151:Connections to class field theory
1151:< 0, the ideal class group of
1089:gave a later proof in 1967. (See
799:Class numbers of quadratic fields
651:) being the functor assigning to
578:The number of ideal classes (the
492:if and only if there is an ideal
323:fundamental theorem of arithmetic
298:, as was recognised at the time.
220:fails in the ring of integers of
2162:which seeks to classify all the
1779:
1766:
1704:
1691:
1672:{\displaystyle (1+{\sqrt {-5}})}
1247:
1203:
1159:
1112:
924:
835:
690:Relation with the group of units
305:was working towards a theory of
34:
1718:, so that the quotient ring of
1354:is not principal, which can be
45:needs additional citations for
2364:Pacific Journal of Mathematics
2339:
2328:
2317:
2091:{\displaystyle b^{2}+5c^{2}=2}
2029:
2023:
1948:
1942:
1909:
1890:
1861:
1855:
1832:
1826:
1666:
1647:
1531:
1525:
1508:{\displaystyle N(uv)=N(u)N(v)}
1502:
1496:
1490:
1484:
1475:
1466:
1414:
1392:
1336:of order 2. Indeed, the ideal
1262:
1251:
1218:
1207:
1174:
1163:
1127:
1116:
939:
928:
850:
839:
772:is a fourth root of 1 (i.e. a
741:Examples of ideal class groups
13:
1:
2355:
1797:were generated by an element
1195:equal to the discriminant of
554:
387:
294:of forms. This gave a finite
2262:formula for the class number
2042:cannot be 2 either, because
1809:would divide both 2 and 1 +
733:is the ideal class group of
573:unique factorization domains
18:Class number (number theory)
7:
2239:
1313:is the ring of integers of
1305:The quadratic integer ring
317:by factorisation using the
264:unique factorization domain
10:
2500:
2426:Algebraische Zahlentheorie
2397:Cambridge University Press
2346:Fröhlich & Taylor 1993
2434:. Vol. 322. Berlin:
2224:is a Galois extension of
916:of algebraic integers of
725:is the group of units of
346:-roots of unity, for any
2310:
2214:is a principal ideal in
2206:is an integral ideal of
1293:case, they are given in
1073:. This result was first
2479:Algebraic number theory
2393:Algebraic number theory
2377:10.2140/pjm.1966.18.219
2160:algebraic number theory
2105:One also computes that
2013:, a contradiction. But
1932:(x) would divide 2. If
1609:{\displaystyle J\neq R}
1356:proved by contradiction
1100:If, on the other hand,
865:quadratic extension of
604:algebraic number theory
370:principal ideal domains
313:in the general case of
230:of the group, which is
2430:
2271:Principal ideal domain
2092:
2036:
2007:
1981:
1961:
1960:{\displaystyle N(x)=1}
1922:
1874:
1873:{\displaystyle N(2)=4}
1839:
1787:
1752:
1732:
1712:
1673:
1634:
1610:
1584:
1564:
1544:
1543:{\displaystyle N(u)=1}
1509:
1450:
1372:
1269:
1225:
1189:binary quadratic forms
1181:
1134:
1067:
966:
946:
910:
890:
889:{\displaystyle d<0}
857:
817:
611:algebraic number field
569:principal ideal domain
527:, or more generally a
525:algebraic number field
448:. (Here the notation (
338:in that group for the
247:The theory extends to
146:algebraic number field
2286:Fermat's Last Theorem
2256:Brauer–Siegel theorem
2198:becomes principal in
2093:
2037:
2008:
1982:
1962:
1923:
1875:
1840:
1788:
1753:
1733:
1713:
1674:
1635:
1611:
1585:
1565:
1545:
1510:
1451:
1373:
1270:
1226:
1182:
1144:with class number 1.
1135:
1091:Stark–Heegner theorem
1068:
967:
947:
911:
891:
858:
818:
787:is a field, then the
360:Somewhat later again
315:Fermat's Last Theorem
2484:Ideals (ring theory)
2305:Arakelov class group
2251:Class number problem
2246:Class number formula
2053:
2048:Diophantine equation
2035:{\displaystyle N(x)}
2017:
1991:
1971:
1936:
1884:
1849:
1838:{\displaystyle N(x)}
1820:
1762:
1742:
1722:
1687:
1644:
1624:
1594:
1574:
1554:
1519:
1460:
1386:
1362:
1243:
1199:
1155:
1108:
1095:class number problem
976:
956:
920:
900:
874:
831:
807:
764:, where ω is a cube
466:equivalence relation
288:Carl Friedrich Gauss
218:unique factorization
54:improve this article
2176:Hilbert class field
2120:The fact that this
2006:{\displaystyle J=R}
825:square-free integer
292:equivalence classes
253:fields of fractions
69:"Ideal class group"
2385:Fröhlich, Albrecht
2300:algebraic geometry
2291:Narrow class group
2276:Algebraic K-theory
2210:then the image of
2186:of a number field
2164:abelian extensions
2156:Class field theory
2088:
2032:
2003:
1977:
1957:
1918:
1870:
1845:would divide both
1835:
1783:
1748:
1728:
1708:
1669:
1630:
1606:
1580:
1560:
1540:
1505:
1456:, which satisfies
1446:
1368:
1265:
1237:narrow class group
1221:
1177:
1130:
1063:
962:
942:
906:
886:
853:
813:
719:group homomorphism
709:Define a map from
638:algebraic K-theory
547:) is a product of
521:algebraic integers
366:algebraic integers
2445:978-3-540-65399-8
2406:978-0-521-43834-6
2168:Galois extensions
2132:6 = 2 × 3 = (1 +
1980:{\displaystyle x}
1907:
1758:is isomorphic to
1751:{\displaystyle J}
1731:{\displaystyle R}
1664:
1640:modulo the ideal
1633:{\displaystyle R}
1583:{\displaystyle R}
1563:{\displaystyle u}
1412:
1371:{\displaystyle R}
1283:quadratic integer
1259:
1215:
1171:
1124:
965:{\displaystyle d}
936:
909:{\displaystyle R}
847:
816:{\displaystyle d}
778:Euclidean domains
774:square root of −1
619:Minkowski's bound
537:ideal class group
406:fractional ideals
325:– to hold in the
307:cyclotomic fields
282:: in the case of
187:fractional ideals
138:ideal class group
130:
129:
122:
104:
16:(Redirected from
2491:
2465:
2433:
2422:Neukirch, Jürgen
2417:
2380:
2379:
2349:
2343:
2337:
2332:
2326:
2321:
2145:
2144:
2138:
2137:
2097:
2095:
2094:
2089:
2081:
2080:
2065:
2064:
2041:
2039:
2038:
2033:
2012:
2010:
2009:
2004:
1986:
1984:
1983:
1978:
1966:
1964:
1963:
1958:
1927:
1925:
1924:
1919:
1908:
1900:
1879:
1877:
1876:
1871:
1844:
1842:
1841:
1836:
1816:. Then the norm
1815:
1814:
1792:
1790:
1789:
1784:
1782:
1774:
1769:
1757:
1755:
1754:
1749:
1737:
1735:
1734:
1729:
1717:
1715:
1714:
1709:
1707:
1699:
1694:
1678:
1676:
1675:
1670:
1665:
1657:
1639:
1637:
1636:
1631:
1615:
1613:
1612:
1607:
1590:. First of all,
1589:
1587:
1586:
1581:
1569:
1567:
1566:
1561:
1549:
1547:
1546:
1541:
1514:
1512:
1511:
1506:
1455:
1453:
1452:
1447:
1445:
1444:
1429:
1428:
1413:
1405:
1377:
1375:
1374:
1369:
1349:
1348:
1323:
1322:
1274:
1272:
1271:
1266:
1260:
1255:
1250:
1230:
1228:
1227:
1222:
1216:
1211:
1206:
1186:
1184:
1183:
1178:
1172:
1167:
1162:
1139:
1137:
1136:
1131:
1125:
1120:
1115:
1072:
1070:
1069:
1064:
971:
969:
968:
963:
951:
949:
948:
943:
937:
932:
927:
915:
913:
912:
907:
895:
893:
892:
887:
862:
860:
859:
854:
848:
843:
838:
822:
820:
819:
814:
584:
583:
567:is from being a
486:identity element
362:Richard Dedekind
249:Dedekind domains
243:
234:, is called the
225:
207:
198:
191:ring of integers
180:
171:
152:
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
2499:
2498:
2494:
2493:
2492:
2490:
2489:
2488:
2469:
2468:
2446:
2436:Springer-Verlag
2407:
2358:
2353:
2352:
2344:
2340:
2333:
2329:
2322:
2318:
2313:
2242:
2158:is a branch of
2153:
2142:
2140:
2135:
2133:
2076:
2072:
2060:
2056:
2054:
2051:
2050:
2018:
2015:
2014:
1992:
1989:
1988:
1972:
1969:
1968:
1937:
1934:
1933:
1899:
1885:
1882:
1881:
1850:
1847:
1846:
1821:
1818:
1817:
1812:
1810:
1778:
1770:
1765:
1763:
1760:
1759:
1743:
1740:
1739:
1723:
1720:
1719:
1703:
1695:
1690:
1688:
1685:
1684:
1656:
1645:
1642:
1641:
1625:
1622:
1621:
1595:
1592:
1591:
1575:
1572:
1571:
1555:
1552:
1551:
1550:if and only if
1520:
1517:
1516:
1461:
1458:
1457:
1440:
1436:
1424:
1420:
1404:
1387:
1384:
1383:
1363:
1360:
1359:
1346:
1344:
1320:
1318:
1303:
1254:
1246:
1244:
1241:
1240:
1210:
1202:
1200:
1197:
1196:
1166:
1158:
1156:
1153:
1152:
1119:
1111:
1109:
1106:
1105:
977:
974:
973:
957:
954:
953:
931:
923:
921:
918:
917:
901:
898:
897:
875:
872:
871:
842:
834:
832:
829:
828:
808:
805:
804:
801:
789:polynomial ring
743:
696:Dedekind domain
692:
661:
646:
581:
580:
557:
529:Dedekind domain
519:is the ring of
454:principal ideal
398:integral domain
390:
382:Dedekind domain
280:quadratic forms
272:
239:
221:
205:
200:
194:
178:
173:
169:
162:
157:
148:
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
2497:
2487:
2486:
2481:
2467:
2466:
2444:
2418:
2405:
2389:Taylor, Martin
2381:
2370:(2): 219–222,
2357:
2354:
2351:
2350:
2338:
2327:
2315:
2314:
2312:
2309:
2308:
2307:
2302:
2293:
2288:
2283:
2278:
2273:
2268:
2263:
2253:
2248:
2241:
2238:
2234:
2233:
2219:
2152:
2149:
2148:
2147:
2087:
2084:
2079:
2075:
2071:
2068:
2063:
2059:
2031:
2028:
2025:
2022:
2002:
1999:
1996:
1987:is a unit and
1976:
1956:
1953:
1950:
1947:
1944:
1941:
1917:
1914:
1911:
1906:
1903:
1898:
1895:
1892:
1889:
1869:
1866:
1863:
1860:
1857:
1854:
1834:
1831:
1828:
1825:
1781:
1777:
1773:
1768:
1747:
1727:
1706:
1702:
1698:
1693:
1668:
1663:
1660:
1655:
1652:
1649:
1629:
1616:, because the
1605:
1602:
1599:
1579:
1559:
1539:
1536:
1533:
1530:
1527:
1524:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1443:
1439:
1435:
1432:
1427:
1423:
1419:
1416:
1411:
1408:
1403:
1400:
1397:
1394:
1391:
1367:
1352:
1351:
1302:
1299:
1264:
1258:
1253:
1249:
1220:
1214:
1209:
1205:
1176:
1170:
1165:
1161:
1129:
1123:
1118:
1114:
1081:and proven by
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
961:
941:
935:
930:
926:
905:
885:
882:
879:
852:
846:
841:
837:
812:
800:
797:
796:
795:
781:
742:
739:
700:group of units
691:
688:
659:
644:
556:
553:
508:may only be a
389:
386:
319:roots of unity
271:
268:
262:the ring is a
260:if and only if
203:
176:
167:
160:
155:quotient group
128:
127:
42:
40:
33:
26:
9:
6:
4:
3:
2:
2496:
2485:
2482:
2480:
2477:
2476:
2474:
2463:
2459:
2455:
2451:
2447:
2441:
2437:
2432:
2427:
2423:
2419:
2416:
2412:
2408:
2402:
2398:
2394:
2390:
2386:
2382:
2378:
2373:
2369:
2365:
2360:
2359:
2347:
2342:
2336:
2335:Neukirch 1999
2331:
2325:
2320:
2316:
2306:
2303:
2301:
2297:
2294:
2292:
2289:
2287:
2284:
2282:
2281:Galois theory
2279:
2277:
2274:
2272:
2269:
2267:
2264:
2261:
2257:
2254:
2252:
2249:
2247:
2244:
2243:
2237:
2231:
2227:
2223:
2220:
2217:
2213:
2209:
2205:
2201:
2197:
2193:
2192:
2191:
2189:
2185:
2181:
2177:
2173:
2170:with abelian
2169:
2165:
2161:
2157:
2131:
2130:
2129:
2127:
2123:
2118:
2116:
2112:
2108:
2103:
2101:
2085:
2082:
2077:
2073:
2069:
2066:
2061:
2057:
2049:
2045:
2026:
2020:
2000:
1997:
1994:
1974:
1954:
1951:
1945:
1939:
1931:
1915:
1912:
1904:
1901:
1896:
1893:
1887:
1867:
1864:
1858:
1852:
1829:
1823:
1808:
1804:
1800:
1796:
1775:
1771:
1745:
1725:
1700:
1696:
1682:
1661:
1658:
1653:
1650:
1627:
1619:
1618:quotient ring
1603:
1600:
1597:
1577:
1570:is a unit in
1557:
1537:
1534:
1528:
1522:
1499:
1493:
1487:
1481:
1478:
1472:
1469:
1463:
1441:
1437:
1433:
1430:
1425:
1421:
1417:
1409:
1406:
1401:
1398:
1395:
1389:
1381:
1365:
1357:
1342:
1339:
1338:
1337:
1335:
1331:
1327:
1316:
1312:
1308:
1298:
1296:
1292:
1288:
1284:
1281:
1276:
1256:
1238:
1234:
1212:
1194:
1190:
1168:
1150:
1145:
1143:
1142:number fields
1121:
1103:
1098:
1096:
1092:
1088:
1084:
1080:
1076:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
982:
979:
959:
933:
903:
883:
880:
877:
869:
868:
844:
826:
810:
793:
790:
786:
782:
779:
775:
771:
767:
763:
762:
757:
756:
751:
750:
745:
744:
738:
736:
732:
728:
724:
720:
716:
712:
707:
705:
701:
697:
687:
685:
681:
677:
673:
669:
665:
658:
654:
650:
643:
639:
635:
631:
626:
624:
620:
616:
612:
607:
605:
601:
597:
593:
589:
585:
576:
574:
570:
566:
562:
552:
550:
546:
542:
538:
534:
533:abelian group
530:
526:
522:
518:
513:
511:
507:
503:
499:
495:
491:
487:
483:
479:
475:
471:
470:ideal classes
467:
463:
459:
455:
451:
447:
443:
439:
435:
431:
427:
423:
419:
415:
411:
407:
404:~ on nonzero
403:
399:
395:
385:
383:
379:
375:
371:
367:
363:
358:
356:
355:regular prime
352:
349:
345:
341:
337:
333:
328:
324:
320:
316:
312:
308:
304:
299:
297:
296:abelian group
293:
289:
285:
281:
277:
267:
265:
261:
258:
254:
250:
245:
242:
237:
233:
229:
224:
219:
215:
211:
206:
197:
192:
188:
184:
179:
170:
163:
156:
151:
147:
143:
139:
135:
124:
121:
113:
110:February 2010
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
2425:
2392:
2367:
2363:
2348:, Theorem 58
2341:
2330:
2324:Claborn 1966
2319:
2296:Picard group
2235:
2229:
2225:
2221:
2215:
2211:
2207:
2203:
2199:
2195:
2187:
2183:
2172:Galois group
2154:
2126:irreducibles
2121:
2119:
2114:
2110:
2106:
2104:
2043:
1929:
1806:
1802:
1798:
1794:
1358:as follows.
1353:
1340:
1329:
1325:
1314:
1310:
1306:
1304:
1295:OEIS A000924
1287:OEIS A003649
1277:
1232:
1193:discriminant
1148:
1146:
1101:
1099:
1087:Harold Stark
1083:Kurt Heegner
866:
802:
791:
784:
769:
760:
754:
747:
734:
726:
714:
710:
708:
693:
683:
679:
675:
671:
667:
663:
656:
652:
648:
641:
629:
627:
615:discriminant
608:
599:
595:
587:
582:class number
579:
577:
564:
560:
558:
549:prime ideals
544:
540:
536:
516:
515:However, if
514:
505:
501:
497:
493:
477:
473:
469:
461:
457:
452:) means the
449:
445:
441:
437:
433:
429:
425:
421:
417:
413:
409:
393:
391:
378:prime ideals
374:proper ideal
359:
350:
348:prime number
343:
331:
303:Ernst Kummer
300:
273:
246:
240:
236:class number
235:
222:
201:
195:
174:
165:
158:
149:
141:
137:
131:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
2202:, i.e., if
1324:). It does
1075:conjectured
482:commutative
432:such that (
400:, define a
142:class group
134:mathematics
2473:Categories
2462:0956.11021
2356:References
2260:asymptotic
2180:unramified
1681:isomorphic
1343:= (2, 1 +
1289:; for the
746:The rings
729:, and its
634:functorial
623:ideal norm
592:isomorphic
555:Properties
496:such that
388:Definition
251:and their
80:newspapers
2139:) × (1 −
1902:−
1659:−
1601:≠
1407:−
1382:function
1291:imaginary
1058:−
1049:−
1040:−
1031:−
1022:−
1013:−
1004:−
995:−
986:−
766:root of 1
678:), where
613:of small
214:principal
144:) of an
2424:(1999).
2391:(1993),
2240:See also
2100:modulo 5
731:cokernel
617:, using
402:relation
210:subgroup
164: /
2454:1697859
2415:1215934
2141:√
2134:√
1811:√
1805:, then
1738:modulo
1345:√
1319:√
640:, with
490:inverse
336:torsion
257:trivial
208:is its
189:of the
181:is the
153:is the
94:scholar
2460:
2452:
2442:
2413:
2403:
1793:. If
1515:, and
1378:has a
1334:cyclic
1231:. For
870:. If
758:, and
723:kernel
721:; its
600:finite
535:, the
523:in an
510:monoid
396:is an
311:proofs
301:Later
284:binary
232:finite
226:. The
199:, and
172:where
136:, the
96:
89:
82:
75:
67:
2311:Notes
2115:other
1967:then
1928:, so
1079:Gauss
863:is a
823:is a
704:units
340:field
327:rings
276:ideal
228:order
183:group
101:JSTOR
87:books
2440:ISBN
2401:ISBN
2258:—an
1880:and
1380:norm
1280:real
1278:For
1147:For
881:<
768:and
666:) =
424:and
140:(or
73:news
2458:Zbl
2372:doi
1801:of
1683:to
1679:is
1620:of
1332:is
1326:not
1239:of
1191:of
1077:by
1061:163
803:If
783:If
586:of
539:of
472:of
456:of
440:= (
428:of
412:by
408:of
392:If
357:).
342:of
238:of
212:of
193:of
185:of
132:In
56:by
2475::
2456:.
2450:MR
2448:.
2438:.
2428:.
2411:MR
2409:,
2399:,
2387:;
2368:18
2366:,
2146:).
2143:−5
2136:−5
2128::
2102:.
1813:−5
1347:−5
1321:−5
1309:=
1297:.
1275:.
1097:.
1052:67
1043:43
1034:19
1025:11
972::
752:,
606:.
551:.
512:.
498:IJ
416:~
266:.
244:.
2464:.
2374::
2232:.
2230:K
2226:K
2222:L
2218:.
2216:L
2212:I
2208:K
2204:I
2200:L
2196:K
2188:K
2184:L
2122:J
2111:J
2107:J
2086:2
2083:=
2078:2
2074:c
2070:5
2067:+
2062:2
2058:b
2044:R
2030:)
2027:x
2024:(
2021:N
2001:R
1998:=
1995:J
1975:x
1955:1
1952:=
1949:)
1946:x
1943:(
1940:N
1930:N
1916:6
1913:=
1910:)
1905:5
1897:+
1894:1
1891:(
1888:N
1868:4
1865:=
1862:)
1859:2
1856:(
1853:N
1833:)
1830:x
1827:(
1824:N
1807:x
1803:R
1799:x
1795:J
1780:Z
1776:3
1772:/
1767:Z
1746:J
1726:R
1705:Z
1701:6
1697:/
1692:Z
1667:)
1662:5
1654:+
1651:1
1648:(
1628:R
1604:R
1598:J
1578:R
1558:u
1538:1
1535:=
1532:)
1529:u
1526:(
1523:N
1503:)
1500:v
1497:(
1494:N
1491:)
1488:u
1485:(
1482:N
1479:=
1476:)
1473:v
1470:u
1467:(
1464:N
1442:2
1438:b
1434:5
1431:+
1426:2
1422:a
1418:=
1415:)
1410:5
1402:b
1399:+
1396:a
1393:(
1390:N
1366:R
1350:)
1341:J
1330:R
1317:(
1315:Q
1311:Z
1307:R
1263:)
1257:d
1252:(
1248:Q
1233:d
1219:)
1213:d
1208:(
1204:Q
1175:)
1169:d
1164:(
1160:Q
1149:d
1128:)
1122:d
1117:(
1113:Q
1102:d
1055:,
1046:,
1037:,
1028:,
1019:,
1016:7
1010:,
1007:3
1001:,
998:2
992:,
989:1
983:=
980:d
960:d
940:)
934:d
929:(
925:Q
904:R
884:0
878:d
867:Q
851:)
845:d
840:(
836:Q
811:d
792:k
785:k
770:i
761:Z
755:Z
749:Z
735:R
727:R
715:R
711:R
684:R
682:(
680:C
676:R
674:(
672:C
670:×
668:Z
664:R
662:(
660:0
657:K
653:R
649:R
647:(
645:0
642:K
630:R
596:R
588:R
565:R
561:R
545:R
541:R
517:R
506:R
502:J
494:J
478:I
474:R
462:a
458:R
450:a
446:J
444:)
442:b
438:I
436:)
434:a
430:R
426:b
422:a
418:J
414:I
410:R
394:R
351:p
344:p
334:-
332:p
241:K
223:K
204:K
202:P
196:K
177:K
175:J
168:K
166:P
161:K
159:J
150:K
123:)
117:(
112:)
108:(
98:·
91:·
84:·
77:·
50:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
