659:
651:
2145:
1233:
1059:
654:
The blue crosses are the number of totally real cubic fields of bounded discriminant. The black line is the asymptotic distribution to first order whereas the green line includes the second order term.
662:
The blue crosses are the number of complex cubic fields of bounded discriminant. The black line is the asymptotic distribution to first order whereas the green line includes the second order term.
356:
145:
1228:{\displaystyle N^{\pm }(X)\sim {\frac {A_{\pm }}{12\zeta (3)}}X+{\frac {4\zeta ({\frac {1}{3}})B_{\pm }}{5\Gamma ({\frac {2}{3}})^{3}\zeta ({\frac {5}{3}})}}X^{\frac {5}{6}}}
869:
1274:
292:
1923:
424:
is an example of a totally real cubic field that is not cyclic. Its discriminant is 148, the smallest discriminant of a non-cyclic totally real cubic field.
358:. This is an example of a pure cubic field, and hence of a complex cubic field. In fact, of all pure cubic fields, it has the smallest discriminant (in
667:
252:
402:
yields a cyclic cubic field, and hence a totally real cubic field. It has the smallest discriminant of all totally real cubic fields, namely 49.
1904:
689:, the discriminant of a cubic field will be positive precisely when the field is totally real, and negative if it is a complex cubic field.
1970:
2165:
581:
255:, then the proportion of cubic fields which are cyclic approaches zero as the bound on the discriminant approaches infinity.
1888:
993:
The study of the number of cubic fields whose discriminant is less than a given bound is a current area of research. Let
2170:
71:
1309:
436:
1733:
This theorem yields the only two proven cases of the Cohen-Lenstra heuristics for class groups of quadratic fields.
1794:
1629:
1034:
1776:. Translations of Mathematical Monographs. Vol. 10. Providence, Rhode Island: American Mathematical Society.
1701:
523:
44:
321:
32:
2149:
658:
1038:
1009:)) denote the number of totally real (respectively complex) cubic fields whose discriminant is bounded by
538:
155:
40:
96:
1281:
601:
1834:; Shankar, Arul; Tsimerman, Jacob (2013), "On the Davenport–Heilbronn theorem and second order terms",
650:
1979:
1853:
1724:
1637:
817:
440:
244:
177:
2091:
Taniguchi, Takashi; Thorne, Frank (2013), "Secondary terms in counting functions for cubic fields",
251:
is totally real. It is a rare occurrence in the sense that if the set of cubic fields is ordered by
2017:(1930), "Arithmetische Theorie der kubischen Zahlkörper auf klassenkörpertheoretischer Grundlage",
634: = −3. This is the case for which the quadratic field contained in the Galois closure of
83:
380:
is not pure. It has the smallest discriminant (in absolute value) of all cubic fields, namely −23.
2126:
2100:
2080:
2054:
2034:
2003:
1942:
1877:
1843:
1714:
1401:
1255:
189:
268:
1900:
1711:
The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders
1689:
1573:
295:
1534:
2110:
2064:
2026:
1987:
1961:
1932:
1861:
1810:
1645:
1603:
1537:
1436:
1014:
428:
224:
158:
2122:
2076:
1999:
1954:
1914:
1873:
1824:
2118:
2072:
1995:
1950:
1910:
1896:
1869:
1831:
1820:
1706:
1697:
1557:
1553:
562:
493:
64:
60:
1983:
1857:
1728:
1641:
1965:
1541:
1440:
1413:
1409:
1405:
1289:
1018:
979:
639:
531:
457:
359:
306:
1666:
Concerning algebraic integers derivable from a root of an equation of the third degree
1650:
1624:
1608:
1592:"Effective determination of the decomposition of the rational primes in a cubic field"
1591:
2159:
2038:
2007:
1881:
2130:
580:
is sometimes considered as the "degenerate" quadratic field of discriminant 1). The
2084:
2014:
1483:, §B.4 contains a table of totally real cubic fields and indicates which are cyclic
883:
469:
240:
232:
2068:
1815:
1041:, combined with a study of the tables of cubic fields compiled by Karim Belabas (
305:. Such fields are always complex cubic fields since each positive number has two
1693:
318:
Adjoining the real cube root of 2 to the rational numbers gives the cubic field
174:
162:
28:
17:
1688:
Their work can also be interpreted as a computation of the average size of the
1400:. These fundamental systems of units can be calculated by means of generalized
1865:
1050:
975:
87:
2114:
1046:
263:
1991:
2144:
1431:
Harvey Cohn computed an asymptotic for the number of cyclic cubic fields (
2030:
1946:
1533:
The exact counts were computed by Michel
Olivier and are available at
2059:
1937:
1801:
Belabas, Karim (1997), "A fast algorithm to compute cubic fields",
1334:
pairs of conjugate complex embeddings is determined by the formula
2105:
2045:
Roberts, David P. (2001), "Density of cubic field discriminants",
1895:, Graduate Texts in Mathematics, vol. 138, Berlin, New York:
1848:
1719:
811:(θ) for some number θ that is a root of an irreducible polynomial
657:
649:
431:
are cubic because the degree of a cyclotomic field is equal to φ(
722:. Formulae are known which calculate the prime decomposition of
1968:(1971), "On the density of discriminants of cubic fields. II",
1548:). The second-order term was conjectured by David P. Roberts (
1296:
using methods based on
Bhargava's earlier work, as well as by
928:
this index formula can be combined with the conductor formula
939:
to obtain a decomposition of the polynomial discriminant Δ =
734:
Unlike quadratic fields, several non-isomorphic cubic fields
1921:
Cohn, Harvey (1954), "The density of abelian cubic fields",
696: > 0 there are only finitely many cubic fields
215:
if it contains all three roots of its generating polynomial
1759:
On a generalization of the algorithm of continued fractions
1700:, and thus constitutes one of the few proven cases of the
681:
is the number of conjugate pairs of complex embeddings of
484:
of degree two as its Galois closure. The Galois group Gal(
1021:
determined the first term of the asymptotic behaviour of
1561:
1293:
1514:
1512:
443:
values except for φ(1) = φ(2) = 1.
1258:
1062:
820:
324:
271:
99:
1276:, according to the totally real or complex case, ζ(
1677:
1668:, Master's Thesis, St. Petersburg, 1894 (Russian).
1545:
1444:
1268:
1227:
1033:goes to infinity). By means of an analysis of the
863:
350:
286:
139:
1893:A Course in Computational Algebraic Number Theory
1774:The theory of irrationalities of the third degree
365:The complex cubic field obtained by adjoining to
1924:Proceedings of the American Mathematical Society
1596:Proceedings of the American Mathematical Society
1292:. Proofs of this formula have been published by
1459:, §B.3 contains a table of complex cubic fields
1408:, which have been interpreted geometrically by
1297:
472:of order three. However, any other cubic field
1443:computed the asymptotic for all cubic fields (
8:
1761:(in Russian). Warsaw: Doctoral Dissertation.
262:if it can be obtained by adjoining the real
1625:"Multiplicities of dihedral discriminants"
1529:
1527:
913:(θ) of θ is then defined by Δ =
731:, and so it can be explicitly calculated.
2104:
2058:
1936:
1847:
1814:
1718:
1649:
1607:
1259:
1257:
1214:
1194:
1182:
1168:
1151:
1134:
1122:
1091:
1085:
1067:
1061:
840:
819:
338:
333:
325:
323:
277:
272:
270:
114:
100:
98:
1294:Bhargava, Shankar & Tsimerman (2013)
924:In the case of a non-cyclic cubic field
351:{\displaystyle \mathbf {Q} ({\sqrt{2}})}
1744:
1549:
1536:. The first-order asymptotic is due to
1424:
1042:
1013:in absolute value. In the early 1970s,
1772:Delone, B. N.; Faddeev, D. K. (1964).
1562:Bhargava, Shankar & Tsimerman 2013
1352:− 1. Hence a totally real cubic field
1518:
1504:
1492:
1480:
1468:
1456:
1300:based on the Shintani zeta function.
630:is a pure cubic field if and only if
7:
1791:Introductory algebraic number theory
1552:) and a proof has been published by
1432:
1396:= 1 has a single fundamental unit ε
1053:a more precise asymptotic formula:
405:The field obtained by adjoining to
223:is a cyclic cubic field if it is a
140:{\displaystyle \mathbf {Q} /(f(x))}
1971:Proceedings of the Royal Society A
1789:Şaban Alaca, Kenneth S. Williams,
1162:
553: ≠ 1 then the Galois closure
537: = 1, in which case the only
510:The discriminant of a cubic field
25:
1651:10.1090/S0025-5718-1992-1122071-3
1609:10.1090/S0002-9939-1983-0687621-6
780: = −1228, 22356,
2143:
978:up to a possible factor 2 or 2.
966:associated with the cubic field
772: = −1836, 3969,
748:may share the same discriminant
638:is the cyclotomic field of cube
326:
101:
1590:Llorente, P.; Nart, E. (1983).
1370:= 0 has two independent units ε
950:into the square of the product
898:. Denoting the discriminant of
864:{\displaystyle f(X)=X^{3}-aX+b}
788: = −3299, 32009, and
247:three. This can only happen if
1678:Davenport & Heilbronn 1971
1546:Davenport & Heilbronn 1971
1445:Davenport & Heilbronn 1971
1204:
1191:
1179:
1165:
1144:
1131:
1110:
1104:
1079:
1073:
830:
824:
796: = −70956, 3054132.
756:of these fields is called the
345:
330:
134:
131:
125:
119:
111:
105:
1:
2069:10.1090/s0025-5718-00-01291-6
1816:10.1090/s0025-5718-97-00846-6
1578:Diophantische Approximationen
1316:of an algebraic number field
1312:, the torsion-free unit rank
1298:Taniguchi & Thorne (2013)
982:gave a method for separating
476:is a non-Galois extension of
301:to the rational number field
1636:(198): 831–847 and S55–S58.
1269:{\displaystyle {\sqrt {3}}}
514:can be written uniquely as
31:, specifically the area of
2187:
2047:Mathematics of Computation
1803:Mathematics of Computation
1795:Cambridge University Press
1378:and a complex cubic field
764:. Some small examples are
506:Associated quadratic field
480:and has a field extension
296:cube-free positive integer
287:{\displaystyle {\sqrt{n}}}
258:A cubic field is called a
196:has a non-real root, then
192:. If, on the other hand,
188:and it is an example of a
2093:Duke Mathematical Journal
2019:Mathematische Zeitschrift
1866:10.1007/s00222-012-0433-0
1702:Cohen–Lenstra conjectures
990:in the square part of Δ.
784: = 4 for
2115:10.1215/00127094-2371752
1836:Inventiones Mathematicae
1310:Dirichlet's unit theorem
576: = 1, the subfield
524:fundamental discriminant
437:Euler's totient function
186:totally real cubic field
2166:Algebraic number theory
1757:Voronoi, G. F. (1896).
962:of the quadratic field
692:Given some real number
492:) is isomorphic to the
90:to a field of the form
33:algebraic number theory
1992:10.1098/rspa.1971.0075
1270:
1229:
1039:Shintani zeta function
865:
666:Since the sign of the
663:
655:
612:. The discriminant of
568:whose discriminant is
460:with Galois group Gal(
439:, which only takes on
352:
288:
141:
41:algebraic number field
1709:; Varma, Ila (2014),
1623:Mayer, D. C. (1992).
1282:Riemann zeta function
1271:
1245: = 1 or 3,
1230:
958:and the discriminant
866:
661:
653:
602:relative discriminant
452:A cyclic cubic field
353:
309:non-real cube roots.
289:
142:
74: = 3, then
2152:at Wikimedia Commons
1556:, Arul Shankar, and
1327:real embeddings and
1256:
1060:
818:
803:will be of the form
760:of the discriminant
468:) isomorphic to the
383:Adjoining a root of
322:
269:
231:, in which case its
97:
2171:Field (mathematics)
1984:1971RSPSA.322..405D
1858:2013InMat.193..439B
1729:2014arXiv1401.5875B
1642:1992MaCom..58..831M
1049:, David P. Roberts
882:are integers. The
792: = 6 for
776: = 3 for
768: = 2 for
700:whose discriminant
202:complex cubic field
2053:(236): 1699–1705,
2031:10.1007/BF01246435
1809:(219): 1213–1237,
1402:continued fraction
1266:
1252: = 1 or
1225:
890:is Δ = 4
861:
670:of a number field
664:
656:
561:contains a unique
502:on three letters.
348:
284:
213:cyclic cubic field
190:totally real field
137:
2148:Media related to
2099:(13): 2451–2508,
1978:(1551): 405–420,
1962:Davenport, Harold
1906:978-3-540-55640-4
1264:
1222:
1208:
1202:
1176:
1142:
1114:
429:cyclotomic fields
343:
282:
16:(Redirected from
2178:
2147:
2133:
2108:
2087:
2062:
2041:
2010:
1957:
1940:
1917:
1884:
1851:
1832:Bhargava, Manjul
1827:
1818:
1778:
1777:
1769:
1763:
1762:
1754:
1748:
1747:, Conjecture 3.1
1742:
1736:
1735:
1722:
1707:Bhargava, Manjul
1686:
1680:
1675:
1669:
1662:
1656:
1655:
1653:
1620:
1614:
1613:
1611:
1587:
1581:
1580:, chapter 4, §5.
1571:
1565:
1538:Harold Davenport
1531:
1522:
1516:
1507:
1502:
1496:
1490:
1484:
1478:
1472:
1466:
1460:
1454:
1448:
1437:Harold Davenport
1429:
1275:
1273:
1272:
1267:
1265:
1260:
1234:
1232:
1231:
1226:
1224:
1223:
1215:
1209:
1207:
1203:
1195:
1187:
1186:
1177:
1169:
1157:
1156:
1155:
1143:
1135:
1123:
1115:
1113:
1096:
1095:
1086:
1072:
1071:
1015:Harold Davenport
1001:) (respectively
870:
868:
867:
862:
845:
844:
799:Any cubic field
423:
397:
379:
357:
355:
354:
349:
344:
342:
334:
329:
293:
291:
290:
285:
283:
281:
273:
260:pure cubic field
225:Galois extension
219:. Equivalently,
159:cubic polynomial
146:
144:
143:
138:
118:
104:
65:rational numbers
21:
18:Pure cubic field
2186:
2185:
2181:
2180:
2179:
2177:
2176:
2175:
2156:
2155:
2140:
2090:
2044:
2013:
1966:Heilbronn, Hans
1960:
1938:10.2307/2031963
1920:
1907:
1897:Springer-Verlag
1887:
1830:
1800:
1786:
1781:
1771:
1770:
1766:
1756:
1755:
1751:
1743:
1739:
1705:
1698:quadratic field
1687:
1683:
1676:
1672:
1664:G. F. Voronoi,
1663:
1659:
1622:
1621:
1617:
1589:
1588:
1584:
1572:
1568:
1558:Jacob Tsimerman
1554:Manjul Bhargava
1532:
1525:
1517:
1510:
1503:
1499:
1491:
1487:
1479:
1475:
1467:
1463:
1455:
1451:
1430:
1426:
1422:
1399:
1395:
1388:
1377:
1373:
1369:
1362:
1351:
1344:
1333:
1326:
1306:
1254:
1253:
1251:
1244:
1210:
1178:
1158:
1147:
1124:
1097:
1087:
1063:
1058:
1057:
894: − 27
836:
816:
815:
746:
740:
730:
717:
708:
680:
674:is (−1), where
648:
563:quadratic field
508:
501:
494:symmetric group
450:
410:
384:
370:
362:), namely −108.
320:
319:
315:
267:
266:
95:
94:
61:field extension
53:
23:
22:
15:
12:
11:
5:
2184:
2182:
2174:
2173:
2168:
2158:
2157:
2154:
2153:
2139:
2138:External links
2136:
2135:
2134:
2088:
2042:
2025:(1): 565–582,
2011:
1958:
1931:(3): 476–477,
1918:
1905:
1885:
1842:(2): 439–499,
1828:
1798:
1785:
1782:
1780:
1779:
1764:
1749:
1737:
1681:
1670:
1657:
1615:
1602:(4): 579–585.
1582:
1566:
1542:Hans Heilbronn
1523:
1508:
1497:
1485:
1473:
1461:
1449:
1441:Hans Heilbronn
1423:
1421:
1418:
1404:algorithms by
1397:
1393:
1386:
1375:
1371:
1367:
1360:
1349:
1342:
1331:
1324:
1305:
1302:
1290:Gamma function
1263:
1249:
1242:
1236:
1235:
1221:
1218:
1213:
1206:
1201:
1198:
1193:
1190:
1185:
1181:
1175:
1172:
1167:
1164:
1161:
1154:
1150:
1146:
1141:
1138:
1133:
1130:
1127:
1121:
1118:
1112:
1109:
1106:
1103:
1100:
1094:
1090:
1084:
1081:
1078:
1075:
1070:
1066:
1019:Hans Heilbronn
980:Georgy Voronoy
872:
871:
860:
857:
854:
851:
848:
843:
839:
835:
832:
829:
826:
823:
744:
738:
726:
718:| ≤
713:
704:
678:
647:
644:
640:roots of unity
532:if and only if
507:
504:
499:
458:Galois closure
449:
448:Galois closure
446:
445:
444:
435:), where φ is
425:
403:
381:
363:
360:absolute value
347:
341:
337:
332:
328:
314:
311:
280:
276:
207:A cubic field
148:
147:
136:
133:
130:
127:
124:
121:
117:
113:
110:
107:
103:
52:
49:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2183:
2172:
2169:
2167:
2164:
2163:
2161:
2151:
2146:
2142:
2141:
2137:
2132:
2128:
2124:
2120:
2116:
2112:
2107:
2102:
2098:
2094:
2089:
2086:
2082:
2078:
2074:
2070:
2066:
2061:
2056:
2052:
2048:
2043:
2040:
2036:
2032:
2028:
2024:
2021:(in German),
2020:
2016:
2015:Hasse, Helmut
2012:
2009:
2005:
2001:
1997:
1993:
1989:
1985:
1981:
1977:
1973:
1972:
1967:
1963:
1959:
1956:
1952:
1948:
1944:
1939:
1934:
1930:
1926:
1925:
1919:
1916:
1912:
1908:
1902:
1898:
1894:
1890:
1886:
1883:
1879:
1875:
1871:
1867:
1863:
1859:
1855:
1850:
1845:
1841:
1837:
1833:
1829:
1826:
1822:
1817:
1812:
1808:
1804:
1799:
1796:
1792:
1788:
1787:
1783:
1775:
1768:
1765:
1760:
1753:
1750:
1746:
1741:
1738:
1734:
1730:
1726:
1721:
1716:
1712:
1708:
1703:
1699:
1695:
1691:
1685:
1682:
1679:
1674:
1671:
1667:
1661:
1658:
1652:
1647:
1643:
1639:
1635:
1632:
1631:
1626:
1619:
1616:
1610:
1605:
1601:
1597:
1593:
1586:
1583:
1579:
1575:
1570:
1567:
1563:
1559:
1555:
1551:
1547:
1543:
1539:
1535:
1530:
1528:
1524:
1520:
1515:
1513:
1509:
1506:
1501:
1498:
1494:
1489:
1486:
1482:
1477:
1474:
1470:
1465:
1462:
1458:
1453:
1450:
1446:
1442:
1438:
1434:
1428:
1425:
1419:
1417:
1415:
1411:
1407:
1403:
1392:
1385:
1381:
1366:
1359:
1355:
1348:
1341:
1337:
1330:
1323:
1319:
1315:
1311:
1308:According to
1303:
1301:
1299:
1295:
1291:
1287:
1283:
1279:
1261:
1248:
1241:
1219:
1216:
1211:
1199:
1196:
1188:
1183:
1173:
1170:
1159:
1152:
1148:
1139:
1136:
1128:
1125:
1119:
1116:
1107:
1101:
1098:
1092:
1088:
1082:
1076:
1068:
1064:
1056:
1055:
1054:
1052:
1048:
1044:
1040:
1036:
1032:
1028:
1024:
1020:
1016:
1012:
1008:
1004:
1000:
996:
991:
989:
985:
981:
977:
973:
969:
965:
961:
957:
953:
949:
946:
942:
938:
935:
931:
927:
922:
920:
916:
912:
909:
905:
901:
897:
893:
889:
885:
881:
877:
858:
855:
852:
849:
846:
841:
837:
833:
827:
821:
814:
813:
812:
810:
807: =
806:
802:
797:
795:
791:
787:
783:
779:
775:
771:
767:
763:
759:
755:
752:. The number
751:
747:
737:
732:
729:
725:
721:
716:
712:
707:
703:
699:
695:
690:
688:
684:
677:
673:
669:
660:
652:
645:
643:
641:
637:
633:
629:
624:
622:
619:
615:
611:
607:
603:
599:
595:
591:
587:
583:
579:
575:
572:(in the case
571:
567:
564:
560:
556:
552:
548:
544:
540:
536:
533:
529:
525:
521:
517:
513:
505:
503:
498:
495:
491:
487:
483:
479:
475:
471:
467:
463:
459:
455:
447:
442:
438:
434:
430:
426:
421:
417:
413:
408:
404:
401:
395:
391:
387:
382:
377:
373:
368:
364:
361:
339:
335:
317:
316:
312:
310:
308:
304:
300:
297:
278:
274:
265:
261:
256:
254:
250:
246:
242:
238:
234:
230:
226:
222:
218:
214:
210:
205:
203:
199:
195:
191:
187:
183:
179:
176:
172:
168:
164:
160:
157:
153:
128:
122:
115:
108:
93:
92:
91:
89:
85:
81:
77:
73:
69:
66:
62:
58:
50:
48:
46:
42:
38:
34:
30:
19:
2096:
2092:
2060:math/9904190
2050:
2046:
2022:
2018:
1975:
1969:
1928:
1922:
1892:
1889:Cohen, Henri
1839:
1835:
1806:
1802:
1790:
1773:
1767:
1758:
1752:
1745:Roberts 2001
1740:
1732:
1710:
1704:: see, e.g.
1692:part of the
1684:
1673:
1665:
1660:
1633:
1628:
1618:
1599:
1595:
1585:
1577:
1574:H. Minkowski
1569:
1550:Roberts 2001
1500:
1488:
1476:
1464:
1452:
1427:
1390:
1383:
1379:
1364:
1357:
1353:
1346:
1339:
1335:
1328:
1321:
1317:
1313:
1307:
1285:
1277:
1246:
1239:
1237:
1043:Belabas 1997
1030:
1026:
1022:
1010:
1006:
1002:
998:
994:
992:
987:
983:
971:
967:
963:
959:
955:
951:
947:
944:
940:
936:
933:
929:
925:
923:
918:
914:
910:
907:
903:
899:
895:
891:
887:
884:discriminant
879:
875:
873:
808:
804:
800:
798:
793:
789:
785:
781:
777:
773:
769:
765:
761:
758:multiplicity
757:
753:
749:
742:
735:
733:
727:
723:
719:
714:
710:
705:
701:
697:
693:
691:
686:
682:
675:
671:
668:discriminant
665:
646:Discriminant
635:
631:
627:
625:
620:
617:
613:
609:
605:
597:
593:
589:
585:
577:
573:
569:
565:
558:
554:
550:
546:
542:
534:
527:
519:
515:
511:
509:
496:
489:
485:
481:
477:
473:
470:cyclic group
465:
461:
453:
451:
432:
419:
415:
411:
406:
399:
393:
389:
385:
375:
371:
366:
302:
298:
259:
257:
253:discriminant
248:
236:
233:Galois group
228:
220:
216:
212:
211:is called a
208:
206:
201:
200:is called a
197:
193:
185:
184:is called a
181:
170:
166:
163:coefficients
151:
149:
79:
78:is called a
75:
67:
56:
54:
36:
26:
2150:Cubic field
1694:class group
1630:Math. Comp.
1051:conjectured
1045:) and some
1029:) (i.e. as
709:satisfies |
549:itself. If
456:is its own
156:irreducible
82:. Any such
80:cubic field
37:cubic field
29:mathematics
2160:Categories
1784:References
1519:Cohen 1993
1505:Hasse 1930
1493:Cohen 1993
1481:Cohen 1993
1469:Cohen 1993
1457:Cohen 1993
1304:Unit group
1047:heuristics
976:squarefree
626:The field
530:is cyclic
409:a root of
369:a root of
173:has three
88:isomorphic
51:Definition
2106:1102.2914
2039:121649559
2008:122814162
1882:253738365
1849:1005.0672
1720:1401.5875
1690:3-torsion
1435:), while
1433:Cohn 1954
1288:) is the
1280:) is the
1189:ζ
1163:Γ
1153:±
1129:ζ
1102:ζ
1093:±
1083:∼
1069:±
847:−
582:conductor
264:cube root
2131:16463250
1891:(1993),
1521:, §6.4.5
1284:, and Γ(
986:(θ) and
970:, where
539:subfield
526:. Then,
313:Examples
2123:3127806
2085:7524750
2077:1836927
2000:0491593
1980:Bibcode
1955:0064076
1947:2031963
1915:1228206
1874:3090184
1854:Bibcode
1825:1415795
1797:, 2004.
1725:Bibcode
1638:Bibcode
1414:Faddeev
1406:Voronoi
1037:of the
1035:residue
741:, ...,
600:is the
307:complex
180:, then
63:of the
47:three.
2129:
2121:
2083:
2075:
2037:
2006:
1998:
1953:
1945:
1913:
1903:
1880:
1872:
1823:
1495:, §B.4
1471:, §B.3
1410:Delone
1238:where
906:, the
874:where
596:, and
518:where
241:cyclic
169:. If
154:is an
150:where
72:degree
45:degree
39:is an
2127:S2CID
2101:arXiv
2081:S2CID
2055:arXiv
2035:S2CID
2004:S2CID
1943:JSTOR
1878:S2CID
1844:arXiv
1715:arXiv
1696:of a
1420:Notes
1382:with
1363:= 3,
1356:with
1320:with
908:index
685:into
608:over
588:over
522:is a
294:of a
245:order
235:over
178:roots
161:with
84:field
59:is a
1901:ISBN
1540:and
1439:and
1412:and
1017:and
878:and
441:even
175:real
35:, a
2111:doi
2097:162
2065:doi
2027:doi
1988:doi
1976:322
1933:doi
1862:doi
1840:193
1811:doi
1646:doi
1604:doi
1374:, ε
974:is
954:(θ)
943:(θ)
917:(θ)
902:by
886:of
616:is
604:of
592:is
584:of
557:of
545:is
541:of
427:No
422:− 1
418:− 3
398:to
396:− 1
392:− 2
378:− 1
243:of
239:is
227:of
165:in
86:is
70:of
55:If
43:of
27:In
2162::
2125:,
2119:MR
2117:,
2109:,
2095:,
2079:,
2073:MR
2071:,
2063:,
2051:70
2049:,
2033:,
2023:31
2002:,
1996:MR
1994:,
1986:,
1974:,
1964:;
1951:MR
1949:,
1941:,
1927:,
1911:MR
1909:,
1899:,
1876:,
1870:MR
1868:,
1860:,
1852:,
1838:,
1821:MR
1819:,
1807:66
1805:,
1793:,
1731:,
1723:,
1713:,
1644:.
1634:58
1627:.
1600:87
1598:.
1594:.
1576:,
1564:).
1526:^
1511:^
1447:).
1416:.
1389:=
1345:+
1338:=
1099:12
932:=
921:.
642:.
623:.
516:df
414:+
388:+
374:+
204:.
2113::
2103::
2067::
2057::
2029::
1990::
1982::
1935::
1929:5
1864::
1856::
1846::
1813::
1727::
1717::
1654:.
1648::
1640::
1612:.
1606::
1560:(
1544:(
1398:1
1394:2
1391:r
1387:1
1384:r
1380:K
1376:2
1372:1
1368:2
1365:r
1361:1
1358:r
1354:K
1350:2
1347:r
1343:1
1340:r
1336:r
1332:2
1329:r
1325:1
1322:r
1318:K
1314:r
1286:s
1278:s
1262:3
1250:±
1247:B
1243:±
1240:A
1220:6
1217:5
1212:X
1205:)
1200:3
1197:5
1192:(
1184:3
1180:)
1174:3
1171:2
1166:(
1160:5
1149:B
1145:)
1140:3
1137:1
1132:(
1126:4
1120:+
1117:X
1111:)
1108:3
1105:(
1089:A
1080:)
1077:X
1074:(
1065:N
1031:X
1027:X
1025:(
1023:N
1011:X
1007:X
1005:(
1003:N
999:X
997:(
995:N
988:f
984:i
972:d
968:K
964:k
960:d
956:f
952:i
948:d
945:f
941:i
937:d
934:f
930:D
926:K
919:D
915:i
911:i
904:D
900:K
896:b
892:a
888:f
880:b
876:a
859:b
856:+
853:X
850:a
842:3
838:X
834:=
831:)
828:X
825:(
822:f
809:Q
805:K
801:K
794:D
790:m
786:D
782:m
778:D
774:m
770:D
766:m
762:D
754:m
750:D
745:m
743:K
739:1
736:K
728:K
724:D
720:N
715:K
711:D
706:K
702:D
698:K
694:N
687:C
683:K
679:2
676:r
672:K
636:K
632:d
628:K
621:f
618:d
614:N
610:K
606:N
598:f
594:f
590:k
586:N
578:Q
574:d
570:d
566:k
559:K
555:N
551:d
547:Q
543:K
535:d
528:K
520:d
512:K
500:3
497:S
490:Q
488:/
486:N
482:N
478:Q
474:K
466:Q
464:/
462:K
454:K
433:n
420:x
416:x
412:x
407:Q
400:Q
394:x
390:x
386:x
376:x
372:x
367:Q
346:)
340:3
336:2
331:(
327:Q
303:Q
299:n
279:3
275:n
249:K
237:Q
229:Q
221:K
217:f
209:K
198:K
194:f
182:K
171:f
167:Q
152:f
135:)
132:)
129:x
126:(
123:f
120:(
116:/
112:]
109:x
106:[
102:Q
76:K
68:Q
57:K
20:)
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