1308:
1281:
The regulator of an algebraic number field of degree greater than 2 is usually quite cumbersome to calculate, though there are now computer algebra packages that can do it in many cases. It is usually much easier to calculate the product
1117:
760:
480:
1530:
1204:
2141:
1485:
923:
354:
994:
1858:
1245:
consisting of all vectors whose entries have sum 0, and by
Dirichlet's unit theorem the image is a lattice in this subspace. The volume of a fundamental domain of this lattice is
1119:
has the property that the sum of any row is zero (because all units have norm 1, and the log of the norm is the sum of the entries in a row). This implies that the absolute value
1243:
818:
1687:
1640:
1574:
1276:
1596:
1335:
967:
945:
583:
542:
511:
413:
268:
1822:
859:
209:
556:, the rank is 1 if it is a real quadratic field, and 0 if an imaginary quadratic field. The theory for real quadratic fields is essentially the theory of
2513:
2383:
707:
2674:
589:
called the regulator. In principle a basis for the units can be effectively computed; in practice the calculations are quite involved when
2433:. Annals of Mathematics Studies. Vol. 74. Princeton, NJ: Princeton University Press and University of Tokyo Press. pp. 36–42.
1299:, and the main difficulty in calculating the class number of an algebraic number field is usually the calculation of the regulator.
17:
2079:
2640:
2598:
2486:
448:
2558:
2438:
2340:
2590:
1449:
366:
316:
1490:
1142:). It measures the "density" of the units: if the regulator is small, this means that there are "lots" of units.
1152:
1455:
43:
864:
2474:
2466:
2332:
322:
1703:
1832:
1289:
1123:
of the determinant of the submatrix formed by deleting one column is independent of the column. The number
629:
is a finite extension of number fields with degree greater than 1 and the units groups for the integers of
2524:
2392:
1213:
777:
2244:
585:
and imaginary quadratic fields, which have rank 0. The 'size' of the units is measured in general by a
1434:
609:
1645:
1248:
2221:
1601:
1535:
1307:
443:
87:
47:
1952:
that plays the same role as the classical regulator does for the group of units, which is a group
1579:
1318:
950:
928:
566:
525:
494:
396:
251:
39:
1989:
1112:{\displaystyle \left(N_{j}\log \left|u_{i}^{(j)}\right|\right)_{i=1,\dots ,r,\;j=1,\dots ,r+1}}
697:
2000:, similar to the classical regulator as a determinant of logarithms of units, attached to any
1966:
838:
2650:
2608:
2496:
2001:
1445:
1296:
2658:
2616:
2568:
2504:
2448:
2350:
1311:
A fundamental domain in logarithmic space of the group of units of the cyclic cubic field
8:
2289:
2184:
2024:
1977:
604:. For a number field with at least one real embedding the torsion must therefore be only
216:
174:
27:
Gives the rank of the group of units in the ring of algebraic integers of a number field
1827:
557:
55:
2624:
2576:
2636:
2594:
2554:
2482:
2434:
2336:
2277:
1938:
69:
1424:. The area of the fundamental domain is approximately 0.910114, so the regulator of
2654:
2612:
2564:
2500:
2444:
2418:
2346:
980:
to 1 or 2 if the corresponding embedding is real or complex respectively. Then the
684:
1879:, is approximately 0.5255. A basis of the group of units modulo roots of unity is
2646:
2604:
2550:
2492:
2478:
2365:
2239:
645:
is a totally complex quadratic extension. The converse holds too. (An example is
553:
1145:
The regulator has the following geometric interpretation. The map taking a unit
168:
51:
1131:
of the algebraic number field (it does not depend on the choice of generators
2668:
2320:
2234:
2188:
755:{\displaystyle \mathbb {Q} \oplus O_{K,S}\otimes _{\mathbb {Z} }\mathbb {Q} }
701:
2020:
1993:
1962:
680:
601:
73:
2549:. Graduate Texts in Mathematics. Vol. 110 (2nd ed.). New York:
586:
212:
31:
2542:
1970:
86:
The statement is that the group of units is finitely generated and has
596:
The torsion in the group of units is the set of all roots of unity of
83:
is a positive real number that determines how "dense" the units are.
1965:
and others. Such higher regulators play a role, for example, in the
1961:. A theory of such regulators has been in development, with work of
1937:
A 'higher' regulator refers to a construction for a function on an
90:(maximal number of multiplicatively independent elements) equal to
689:
653:
equal to an imaginary quadratic field; both have unit rank 0.)
163:
is based on the idea that there will be as many ways to embed
1437:, or of the rational integers, is 1 (as the determinant of a
2514:"The Life and Work of Gustav Lejeune Dirichlet (1805–1859)"
619:
Totally real fields are special with respect to units. If
2623:
2295:
2331:. CRM Monograph Series. Vol. 11. Providence, RI:
1969:, and are expected to occur in evaluations of certain
1648:
1604:
1538:
1493:
1155:
2082:
1835:
1706:
1582:
1532:. This can be seen as follows. A fundamental unit is
1458:
1321:
1251:
1216:
997:
953:
931:
867:
841:
780:
710:
569:
528:
497:
451:
399:
325:
254:
177:
475:{\displaystyle K\otimes _{\mathbb {Q} }\mathbb {R} }
563:The rank is positive for all number fields besides
2219:-adic logarithms of the generators of this group.
2167:via the diagonal embedding of the global units in
2135:
1852:
1816:
1681:
1634:
1590:
1568:
1524:
1479:
1329:
1270:
1237:
1198:
1111:
961:
939:
917:
853:
812:
754:
656:The theorem not only applies to the maximal order
577:
536:
505:
474:
407:
348:
262:
203:
2471:A Course in Computational Algebraic Number Theory
2364:Prasad, Dipendra; Yogonanda, C. S. (2007-02-23).
679:There is a generalisation of the unit theorem by
2666:
383:the number that are complex; in other words, if
2363:
2215:is the determinant of the matrix formed by the
1576:, and its images under the two embeddings into
139:number of conjugate pairs of complex embeddings
2329:-theory, and zeta functions of elliptic curves
820:are a set of generators for the unit group of
2633:Grundlehren der Mathematischen Wissenschaften
2586:Grundlehren der mathematischen Wissenschaften
2627:; Schmidt, Alexander; Wingberg, Kay (2000),
2584:
1976:at integer values of the argument. See also
1525:{\textstyle \log {\frac {{\sqrt {5}}+1}{2}}}
696:, determining the rank of the unit group in
687:) to describe the structure of the group of
612:, having no real embeddings which also have
608:. There are number fields, for example most
2275:
1199:{\textstyle N_{j}\log \left|u^{(j)}\right|}
433:is the number of non-real complex roots of
2635:, vol. 323, Berlin: Springer-Verlag,
2225:states that this determinant is non-zero.
2136:{\displaystyle U_{1}=\prod _{P|p}U_{1,P}.}
2065:denote the subgroup of principal units in
1079:
2366:A Report on Artin's holomorphy conjecture
1837:
1584:
1480:{\displaystyle \mathbb {Q} ({\sqrt {5}})}
1460:
1323:
1219:
955:
933:
748:
741:
712:
571:
530:
499:
468:
461:
401:
333:
256:
194:
2575:
2511:
2381:
2263:
1306:
918:{\displaystyle u^{(1)},\dots ,u^{(r+1)}}
439:(which come in complex conjugate pairs);
2417:
2187:subgroup of the global units, it is an
349:{\displaystyle K=\mathbb {Q} (\alpha )}
14:
2667:
2402:
2465:
2319:
2307:
2296:Neukirch, Schmidt & Wingberg 2000
1853:{\displaystyle \mathbb {Q} (\alpha )}
1375:, then a set of fundamental units is
824:modulo roots of unity. There will be
2541:
2313:
1932:
482:as a product of fields, there being
215:, or pairs of embeddings related by
2675:Theorems in algebraic number theory
2477:. Vol. 138. Berlin, New York:
2007:
616:for the torsion of its unit group.
24:
1983:
1238:{\displaystyle \mathbb {R} ^{r+1}}
925:for the different embeddings into
813:{\displaystyle u_{1},\dots ,u_{r}}
25:
2686:
2408:Neukirch et al. (2008) p. 626–627
1996:to define what is now called the
2034:above some fixed rational prime
765:
700:of rings of integers. Also, the
424:is the number of real roots and
211:; these will either be into the
2154:denote the set of global units
1757:
1682:{\textstyle (-{\sqrt {5}}+1)/2}
2411:
2375:
2357:
2301:
2269:
2257:
2105:
1847:
1841:
1668:
1649:
1635:{\textstyle ({\sqrt {5}}+1)/2}
1621:
1605:
1569:{\textstyle ({\sqrt {5}}+1)/2}
1555:
1539:
1474:
1464:
1271:{\displaystyle R{\sqrt {r+1}}}
1187:
1181:
1040:
1034:
910:
898:
879:
873:
835:, either real or complex. For
343:
337:
198:
184:
44:Peter Gustav Lejeune Dirichlet
13:
1:
2475:Graduate Texts in Mathematics
2459:
2333:American Mathematical Society
2325:Higher regulators, algebraic
389:is the minimal polynomial of
2521:Clay Mathematics Proceedings
1591:{\displaystyle \mathbb {R} }
1330:{\displaystyle \mathbb {Q} }
1295:and the regulator using the
962:{\displaystyle \mathbb {C} }
940:{\displaystyle \mathbb {R} }
578:{\displaystyle \mathbb {Q} }
537:{\displaystyle \mathbb {C} }
506:{\displaystyle \mathbb {R} }
408:{\displaystyle \mathbb {Q} }
263:{\displaystyle \mathbb {Q} }
7:
2629:Cohomology of Number Fields
2228:
1302:
1149:to the vector with entries
649:equal to the rationals and
145:. This characterisation of
10:
2691:
2581:Algebraische Zahlentheorie
2298:, proposition VIII.8.6.11.
2049:denote the local units at
1428:is approximately 0.525455.
610:imaginary quadratic fields
293:Other ways of determining
2589:. Vol. 322. Berlin:
2512:Elstrodt, Jürgen (2007).
1435:imaginary quadratic field
1315:obtained by adjoining to
1210:-dimensional subspace of
126:number of real embeddings
2391:(Thesis). Archived from
2382:Dasgupta, Samit (1999).
2276:Stevenhagen, P. (2012).
2250:
1448:is the logarithm of its
637:have the same rank then
444:tensor product of fields
36:Dirichlet's unit theorem
2547:Algebraic number theory
2245:Shintani's unit theorem
1452:: for example, that of
40:algebraic number theory
18:Regulator (mathematics)
2585:
2137:
1854:
1818:
1817:{\displaystyle \left.}
1683:
1636:
1592:
1570:
1526:
1481:
1429:
1331:
1272:
1239:
1200:
1113:
963:
941:
919:
855:
854:{\displaystyle u\in K}
831:Archimedean places of
814:
774:is a number field and
756:
600:, which form a finite
579:
538:
507:
476:
409:
350:
264:
205:
2222:Leopoldt's conjecture
2138:
1967:Beilinson conjectures
1855:
1826:The regulator of the
1819:
1684:
1637:
1593:
1571:
1527:
1482:
1332:
1310:
1273:
1240:
1201:
1114:
964:
942:
920:
856:
815:
762:has been determined.
757:
580:
539:
508:
477:
410:
351:
265:
206:
38:is a basic result in
2080:
2002:Artin representation
1833:
1704:
1646:
1602:
1580:
1536:
1491:
1456:
1446:real quadratic field
1433:The regulator of an
1319:
1297:class number formula
1249:
1214:
1206:has an image in the
1153:
995:
951:
929:
865:
839:
778:
708:
641:is totally real and
567:
526:
495:
449:
397:
323:
252:
175:
171:field as the degree
46:. It determines the
2385:Stark's Conjectures
1990:Stark's conjectures
1988:The formulation of
1978:Beilinson regulator
1444:The regulator of a
1044:
217:complex conjugation
2133:
2113:
1850:
1828:cyclic cubic field
1814:
1679:
1632:
1588:
1566:
1522:
1477:
1430:
1364:denotes a root of
1327:
1268:
1235:
1196:
1109:
1024:
959:
937:
915:
851:
810:
752:
575:
548:As an example, if
534:
503:
472:
405:
346:
260:
204:{\displaystyle n=}
201:
70:algebraic integers
2642:978-3-540-66671-4
2600:978-3-540-65399-8
2488:978-3-540-55640-4
2419:Iwasawa, Kenkichi
2321:Bloch, Spencer J.
2096:
1933:Higher regulators
1805:
1797:
1785:
1748:
1736:
1660:
1613:
1547:
1520:
1508:
1472:
1266:
663:but to any order
365:is the number of
319:theorem to write
317:primitive element
16:(Redirected from
2682:
2661:
2625:Neukirch, Jürgen
2620:
2588:
2577:Neukirch, Jürgen
2572:
2538:
2536:
2535:
2529:
2523:. Archived from
2518:
2508:
2453:
2452:
2430:
2426:
2415:
2409:
2406:
2400:
2399:
2397:
2390:
2379:
2373:
2372:
2370:
2361:
2355:
2354:
2328:
2317:
2311:
2305:
2299:
2293:
2287:
2286:
2284:
2273:
2267:
2261:
2218:
2212:
2207:
2182:
2170:
2166:
2157:
2153:
2142:
2140:
2139:
2134:
2129:
2128:
2112:
2108:
2092:
2091:
2075:
2064:
2052:
2048:
2037:
2033:
2029:
2018:
2010:
1973:
1960:
1951:
1942:
1927:
1914:
1896:
1878:
1863:
1859:
1857:
1856:
1851:
1840:
1823:
1821:
1820:
1815:
1810:
1806:
1803:
1802:
1798:
1793:
1786:
1781:
1775:
1753:
1749:
1744:
1737:
1732:
1729:
1699:
1688:
1686:
1685:
1680:
1675:
1661:
1656:
1641:
1639:
1638:
1633:
1628:
1614:
1609:
1597:
1595:
1594:
1589:
1587:
1575:
1573:
1572:
1567:
1562:
1548:
1543:
1531:
1529:
1528:
1523:
1521:
1516:
1509:
1504:
1501:
1486:
1484:
1483:
1478:
1473:
1468:
1463:
1450:fundamental unit
1440:
1427:
1423:
1410:
1392:
1374:
1363:
1359:
1336:
1334:
1333:
1328:
1326:
1314:
1294:
1287:
1277:
1275:
1274:
1269:
1267:
1256:
1244:
1242:
1241:
1236:
1234:
1233:
1222:
1209:
1205:
1203:
1202:
1197:
1195:
1191:
1190:
1165:
1164:
1148:
1141:
1126:
1122:
1118:
1116:
1115:
1110:
1108:
1107:
1053:
1049:
1048:
1043:
1032:
1013:
1012:
990:
979:
968:
966:
965:
960:
958:
946:
944:
943:
938:
936:
924:
922:
921:
916:
914:
913:
883:
882:
860:
858:
857:
852:
830:
819:
817:
816:
811:
809:
808:
790:
789:
761:
759:
758:
753:
751:
746:
745:
744:
734:
733:
715:
692:
685:Claude Chevalley
675:
662:
652:
648:
644:
640:
636:
632:
628:
615:
607:
599:
592:
584:
582:
581:
576:
574:
551:
543:
541:
540:
535:
533:
521:
512:
510:
509:
504:
502:
490:
481:
479:
478:
473:
471:
466:
465:
464:
438:
432:
423:
414:
412:
411:
406:
404:
392:
388:
382:
372:
364:
355:
353:
352:
347:
336:
310:
301:
289:
279:
269:
267:
266:
261:
259:
247:
240:
210:
208:
207:
202:
197:
166:
162:
153:
144:
136:
123:
112:
78:
67:
21:
2690:
2689:
2685:
2684:
2683:
2681:
2680:
2679:
2665:
2664:
2643:
2601:
2591:Springer-Verlag
2561:
2551:Springer-Verlag
2533:
2531:
2527:
2516:
2489:
2479:Springer-Verlag
2462:
2457:
2456:
2441:
2428:
2424:
2416:
2412:
2407:
2403:
2395:
2388:
2380:
2376:
2368:
2362:
2358:
2343:
2326:
2318:
2314:
2306:
2302:
2294:
2290:
2282:
2274:
2270:
2262:
2258:
2253:
2240:Cyclotomic unit
2231:
2216:
2213:-adic regulator
2210:
2205:
2198:
2192:
2181:
2175:
2168:
2165:
2159:
2155:
2152:
2146:
2118:
2114:
2104:
2100:
2087:
2083:
2081:
2078:
2077:
2074:
2066:
2063:
2054:
2050:
2047:
2039:
2035:
2031:
2027:
2016:
2013:
2011:-adic regulator
2008:
1998:Stark regulator
1986:
1984:Stark regulator
1971:
1959:
1953:
1946:
1940:
1935:
1922:
1916:
1904:
1898:
1894:
1887:
1880:
1865:
1861:
1836:
1834:
1831:
1830:
1780:
1776:
1774:
1770:
1731:
1730:
1728:
1724:
1711:
1707:
1705:
1702:
1701:
1690:
1671:
1655:
1647:
1644:
1643:
1624:
1608:
1603:
1600:
1599:
1583:
1581:
1578:
1577:
1558:
1542:
1537:
1534:
1533:
1503:
1502:
1500:
1492:
1489:
1488:
1467:
1459:
1457:
1454:
1453:
1438:
1425:
1418:
1412:
1400:
1394:
1390:
1383:
1376:
1365:
1361:
1338:
1322:
1320:
1317:
1316:
1312:
1305:
1292:
1283:
1255:
1250:
1247:
1246:
1223:
1218:
1217:
1215:
1212:
1211:
1207:
1180:
1176:
1172:
1160:
1156:
1154:
1151:
1150:
1146:
1140:
1132:
1124:
1120:
1054:
1033:
1028:
1020:
1008:
1004:
1003:
999:
998:
996:
993:
992:
981:
978:
970:
954:
952:
949:
948:
932:
930:
927:
926:
897:
893:
872:
868:
866:
863:
862:
840:
837:
836:
825:
804:
800:
785:
781:
779:
776:
775:
768:
747:
740:
739:
735:
723:
719:
711:
709:
706:
705:
690:
674:
664:
661:
657:
650:
646:
642:
638:
634:
630:
620:
613:
605:
597:
590:
570:
568:
565:
564:
558:Pell's equation
554:quadratic field
549:
529:
527:
524:
523:
520:
514:
498:
496:
493:
492:
489:
483:
467:
460:
459:
455:
450:
447:
446:
434:
431:
425:
422:
416:
400:
398:
395:
394:
390:
384:
381:
374:
373:that are real,
370:
363:
357:
332:
324:
321:
320:
309:
303:
300:
294:
287:
281:
277:
271:
255:
253:
250:
249:
248:is Galois over
245:
242:
239:
232:
222:
193:
176:
173:
172:
164:
161:
155:
152:
146:
142:
135:
129:
122:
116:
113:
110:
103:
93:
76:
66:
58:
28:
23:
22:
15:
12:
11:
5:
2688:
2678:
2677:
2663:
2662:
2641:
2621:
2599:
2573:
2559:
2539:
2509:
2487:
2461:
2458:
2455:
2454:
2439:
2410:
2401:
2398:on 2008-05-10.
2374:
2356:
2341:
2312:
2300:
2288:
2268:
2255:
2254:
2252:
2249:
2248:
2247:
2242:
2237:
2230:
2227:
2203:
2196:
2179:
2163:
2150:
2132:
2127:
2124:
2121:
2117:
2111:
2107:
2103:
2099:
2095:
2090:
2086:
2070:
2058:
2043:
2012:
2006:
1985:
1982:
1957:
1934:
1931:
1930:
1929:
1920:
1902:
1892:
1885:
1849:
1846:
1843:
1839:
1824:
1813:
1809:
1801:
1796:
1792:
1789:
1784:
1779:
1773:
1769:
1766:
1763:
1760:
1756:
1752:
1747:
1743:
1740:
1735:
1727:
1723:
1720:
1717:
1714:
1710:
1678:
1674:
1670:
1667:
1664:
1659:
1654:
1651:
1631:
1627:
1623:
1620:
1617:
1612:
1607:
1586:
1565:
1561:
1557:
1554:
1551:
1546:
1541:
1519:
1515:
1512:
1507:
1499:
1496:
1476:
1471:
1466:
1462:
1442:
1416:
1398:
1388:
1381:
1325:
1304:
1301:
1265:
1262:
1259:
1254:
1232:
1229:
1226:
1221:
1194:
1189:
1186:
1183:
1179:
1175:
1171:
1168:
1163:
1159:
1136:
1127:is called the
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1052:
1047:
1042:
1039:
1036:
1031:
1027:
1023:
1019:
1016:
1011:
1007:
1002:
974:
957:
935:
912:
909:
906:
903:
900:
896:
892:
889:
886:
881:
878:
875:
871:
850:
847:
844:
807:
803:
799:
796:
793:
788:
784:
767:
764:
750:
743:
738:
732:
729:
726:
722:
718:
714:
672:
659:
573:
546:
545:
532:
518:
501:
487:
470:
463:
458:
454:
440:
429:
420:
403:
379:
361:
345:
342:
339:
335:
331:
328:
307:
298:
285:
275:
258:
237:
230:
221:
200:
196:
192:
189:
186:
183:
180:
169:complex number
159:
150:
133:
120:
108:
101:
92:
62:
52:group of units
26:
9:
6:
4:
3:
2:
2687:
2676:
2673:
2672:
2670:
2660:
2656:
2652:
2648:
2644:
2638:
2634:
2630:
2626:
2622:
2618:
2614:
2610:
2606:
2602:
2596:
2592:
2587:
2582:
2578:
2574:
2570:
2566:
2562:
2560:0-387-94225-4
2556:
2552:
2548:
2544:
2540:
2530:on 2021-05-22
2526:
2522:
2515:
2510:
2506:
2502:
2498:
2494:
2490:
2484:
2480:
2476:
2472:
2468:
2464:
2463:
2450:
2446:
2442:
2440:0-691-08112-3
2436:
2432:
2420:
2414:
2405:
2394:
2387:
2386:
2378:
2367:
2360:
2352:
2348:
2344:
2342:0-8218-2114-8
2338:
2334:
2330:
2322:
2316:
2309:
2304:
2297:
2292:
2285:. p. 57.
2281:
2280:
2272:
2265:
2264:Elstrodt 2007
2260:
2256:
2246:
2243:
2241:
2238:
2236:
2235:Elliptic unit
2233:
2232:
2226:
2224:
2223:
2214:
2202:
2195:
2190:
2189:abelian group
2186:
2178:
2172:
2162:
2149:
2143:
2130:
2125:
2122:
2119:
2115:
2109:
2101:
2097:
2093:
2088:
2084:
2073:
2069:
2062:
2057:
2046:
2042:
2026:
2023:and for each
2022:
2005:
2003:
1999:
1995:
1991:
1981:
1979:
1975:
1968:
1964:
1956:
1949:
1944:
1926:
1919:
1912:
1908:
1901:
1891:
1884:
1876:
1872:
1868:
1864:is a root of
1844:
1829:
1825:
1811:
1807:
1799:
1794:
1790:
1787:
1782:
1777:
1771:
1767:
1764:
1761:
1758:
1754:
1750:
1745:
1741:
1738:
1733:
1725:
1721:
1718:
1715:
1712:
1708:
1697:
1693:
1676:
1672:
1665:
1662:
1657:
1652:
1629:
1625:
1618:
1615:
1610:
1563:
1559:
1552:
1549:
1544:
1517:
1513:
1510:
1505:
1497:
1494:
1469:
1451:
1447:
1443:
1441:matrix is 1).
1436:
1432:
1431:
1422:
1415:
1408:
1404:
1397:
1387:
1380:
1372:
1368:
1357:
1353:
1349:
1345:
1341:
1309:
1300:
1298:
1291:
1286:
1279:
1263:
1260:
1257:
1252:
1230:
1227:
1224:
1192:
1184:
1177:
1173:
1169:
1166:
1161:
1157:
1143:
1139:
1135:
1130:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1050:
1045:
1037:
1029:
1025:
1021:
1017:
1014:
1009:
1005:
1000:
988:
984:
977:
973:
907:
904:
901:
894:
890:
887:
884:
876:
869:
848:
845:
842:
834:
828:
823:
805:
801:
797:
794:
791:
786:
782:
773:
770:Suppose that
766:The regulator
763:
736:
730:
727:
724:
720:
716:
704:structure of
703:
702:Galois module
699:
698:localizations
695:
694:
686:
682:
677:
671:
667:
654:
627:
623:
617:
611:
603:
594:
588:
561:
559:
555:
517:
486:
456:
452:
445:
441:
437:
428:
419:
387:
378:
368:
360:
340:
329:
326:
318:
314:
313:
312:
306:
297:
291:
284:
274:
244:Note that if
236:
229:
225:
220:
218:
214:
190:
187:
181:
178:
170:
158:
149:
140:
132:
127:
119:
107:
100:
96:
91:
89:
84:
82:
75:
71:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
2632:
2628:
2580:
2546:
2532:. Retrieved
2525:the original
2520:
2470:
2467:Cohen, Henri
2423:Lectures on
2422:
2413:
2404:
2393:the original
2384:
2377:
2359:
2324:
2315:
2303:
2291:
2279:Number Rings
2278:
2271:
2259:
2220:
2209:
2200:
2193:
2183:is a finite-
2176:
2173:
2160:
2158:that map to
2147:
2144:
2071:
2067:
2060:
2055:
2044:
2040:
2021:number field
2014:
1997:
1994:Harold Stark
1987:
1963:Armand Borel
1954:
1947:
1936:
1924:
1917:
1910:
1906:
1899:
1889:
1882:
1874:
1870:
1866:
1695:
1691:
1420:
1413:
1406:
1402:
1395:
1385:
1378:
1370:
1366:
1355:
1351:
1347:
1343:
1339:
1290:class number
1284:
1280:
1144:
1137:
1133:
1128:
986:
982:
975:
971:
832:
826:
821:
771:
769:
688:
681:Helmut Hasse
678:
669:
665:
655:
625:
621:
618:
602:cyclic group
595:
562:
547:
515:
484:
435:
426:
417:
385:
376:
358:
304:
295:
292:
282:
272:
270:then either
243:
234:
227:
223:
213:real numbers
156:
147:
138:
130:
125:
117:
114:
105:
98:
94:
85:
80:
74:number field
63:
59:
35:
29:
2543:Lang, Serge
2310:, Table B.4
1945:with index
683:(and later
587:determinant
356:, and then
32:mathematics
2659:0948.11001
2617:0956.11021
2569:0811.11001
2534:2010-06-13
2505:0786.11071
2460:References
2449:0236.12001
2431:-functions
2351:0958.19001
2308:Cohen 1993
1974:-functions
1939:algebraic
1700:matrix is
1337:a root of
593:is large.
522:copies of
491:copies of
442:write the
367:conjugates
219:, so that
2371:(Report).
2145:Then let
2098:∏
1845:α
1778:−
1768:
1762:×
1722:
1716:×
1689:. So the
1653:−
1498:
1170:
1129:regulator
1093:…
1068:…
1018:
888:…
846:∈
795:…
737:⊗
717:⊕
457:⊗
341:α
81:regulator
2669:Category
2579:(1999).
2545:(1994).
2469:(1993).
2421:(1972).
2323:(2000).
2229:See also
2191:of rank
2053:and let
1860:, where
1393:, where
1303:Examples
969:and set
861:, write
315:use the
2651:1737196
2609:1697859
2497:1228206
1288:of the
415:, then
167:in the
124:is the
54:in the
50:of the
42:due to
2657:
2649:
2639:
2615:
2607:
2597:
2567:
2557:
2503:
2495:
2485:
2447:
2437:
2427:-adic
2349:
2339:
2266:, §8.D
2208:. The
2174:Since
2076:. Set
2038:, let
1950:> 1
1943:-group
1923:= 2 −
1897:where
1804:
1419:= 2 −
991:matrix
693:-units
614:{1,−1}
606:{1,−1}
115:where
79:. The
2528:(PDF)
2517:(PDF)
2396:(PDF)
2389:(PDF)
2369:(PDF)
2283:(PDF)
2251:Notes
2185:index
2025:prime
2019:be a
1439:0 × 0
1360:. If
552:is a
393:over
72:of a
2637:ISBN
2595:ISBN
2555:ISBN
2483:ISBN
2435:ISBN
2337:ISBN
2015:Let
1992:led
1915:and
1698:+ 1)
1642:and
1598:are
1411:and
1346:) =
989:+ 1)
633:and
513:and
311:are
302:and
154:and
137:the
128:and
88:rank
56:ring
48:rank
2655:Zbl
2613:Zbl
2565:Zbl
2501:Zbl
2445:Zbl
2347:Zbl
2206:− 1
2030:of
1913:− 1
1877:− 1
1873:− 2
1765:log
1719:log
1694:× (
1495:log
1487:is
1409:− 1
1358:− 1
1354:− 2
1167:log
1015:log
985:× (
947:or
829:+ 1
369:of
288:= 0
280:or
278:= 0
233:+ 2
141:of
111:− 1
68:of
30:In
2671::
2653:,
2647:MR
2645:,
2631:,
2611:.
2605:MR
2603:.
2593:.
2583:.
2563:.
2553:.
2519:.
2499:.
2493:MR
2491:.
2481:.
2473:.
2443:.
2345:.
2335:.
2199:+
2171:.
2059:1,
2004:.
1980:.
1909:+
1905:=
1888:,
1869:+
1405:+
1401:=
1384:,
1350:+
1285:hR
1278:.
676:.
668:⊂
560:.
427:2r
290:.
226:=
104:+
97:=
34:,
2619:.
2571:.
2537:.
2507:.
2451:.
2429:L
2425:p
2353:.
2327:K
2217:p
2211:p
2204:2
2201:r
2197:1
2194:r
2180:1
2177:E
2169:E
2164:1
2161:U
2156:ε
2151:1
2148:E
2131:.
2126:P
2123:,
2120:1
2116:U
2110:p
2106:|
2102:P
2094:=
2089:1
2085:U
2072:P
2068:U
2061:P
2056:U
2051:P
2045:P
2041:U
2036:p
2032:K
2028:P
2017:K
2009:p
1972:L
1958:1
1955:K
1948:n
1941:K
1928:.
1925:α
1921:2
1918:ε
1911:α
1907:α
1903:1
1900:ε
1895:}
1893:2
1890:ε
1886:1
1883:ε
1881:{
1875:x
1871:x
1867:x
1862:α
1848:)
1842:(
1838:Q
1812:.
1808:]
1800:|
1795:2
1791:1
1788:+
1783:5
1772:|
1759:1
1755:,
1751:|
1746:2
1742:1
1739:+
1734:5
1726:|
1713:1
1709:[
1696:r
1692:r
1677:2
1673:/
1669:)
1666:1
1663:+
1658:5
1650:(
1630:2
1626:/
1622:)
1619:1
1616:+
1611:5
1606:(
1585:R
1564:2
1560:/
1556:)
1553:1
1550:+
1545:5
1540:(
1518:2
1514:1
1511:+
1506:5
1475:)
1470:5
1465:(
1461:Q
1426:K
1421:α
1417:2
1414:ε
1407:α
1403:α
1399:1
1396:ε
1391:}
1389:2
1386:ε
1382:1
1379:ε
1377:{
1373:)
1371:x
1369:(
1367:f
1362:α
1356:x
1352:x
1348:x
1344:x
1342:(
1340:f
1324:Q
1313:K
1293:h
1264:1
1261:+
1258:r
1253:R
1231:1
1228:+
1225:r
1220:R
1208:r
1193:|
1188:)
1185:j
1182:(
1178:u
1174:|
1162:j
1158:N
1147:u
1138:i
1134:u
1125:R
1121:R
1105:1
1102:+
1099:r
1096:,
1090:,
1087:1
1084:=
1081:j
1077:,
1074:r
1071:,
1065:,
1062:1
1059:=
1056:i
1051:)
1046:|
1041:)
1038:j
1035:(
1030:i
1026:u
1022:|
1010:j
1006:N
1001:(
987:r
983:r
976:j
972:N
956:C
934:R
911:)
908:1
905:+
902:r
899:(
895:u
891:,
885:,
880:)
877:1
874:(
870:u
849:K
843:u
833:K
827:r
822:K
806:r
802:u
798:,
792:,
787:1
783:u
772:K
749:Q
742:Z
731:S
728:,
725:K
721:O
713:Q
691:S
673:K
670:O
666:O
660:K
658:O
651:L
647:K
643:L
639:K
635:K
631:L
626:K
624:/
622:L
598:K
591:n
572:Q
550:K
544:.
531:C
519:2
516:r
500:R
488:1
485:r
469:R
462:Q
453:K
436:f
430:2
421:1
418:r
402:Q
391:α
386:f
380:2
377:r
375:2
371:α
362:1
359:r
344:)
338:(
334:Q
330:=
327:K
308:2
305:r
299:1
296:r
286:2
283:r
276:1
273:r
257:Q
246:K
241:.
238:2
235:r
231:1
228:r
224:n
199:]
195:Q
191::
188:K
185:[
182:=
179:n
165:K
160:2
157:r
151:1
148:r
143:K
134:2
131:r
121:1
118:r
109:2
106:r
102:1
99:r
95:r
77:K
64:K
60:O
20:)
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