1767:
1476:
448:, obtained from the field of rational numbers by adjoining a primitive root of unity of a given order. For example, the ordinary integer primes group into four classes, each with probability 1/4, according to their pattern of splitting in the ring of integers corresponding to the 8th roots of unity. In this case, the field extension has degree 4 and is
1762:{\displaystyle \sum _{p\leq x,p\not \mid {\mathfrak {f}}(\rho )}\chi _{\rho }({\text{Fr}}_{p})\log p=rx+O{\biggl (}{\frac {x^{\beta }}{\beta }}+x\exp {\biggl (}{\frac {-c(dn)^{-4}\log x}{3\log {\mathfrak {f}}(\rho )+{\sqrt {\log x}}}}{\biggr )}(dn\log(x{\mathfrak {f}}(\rho )){\biggr )},}
1123:
2110:
The
Chebotarev density theorem reduces the problem of classifying Galois extensions of a number field to that of describing the splitting of primes in extensions. Specifically, it implies that as a Galois extension of
976:
388:
From this description, it appears that as one considers larger and larger primes, the frequency of a prime splitting completely approaches 1/2, and likewise for the primes that remain primes in
436:
2076:
1306:
821:
is abelian the classes of course each have size 1. For the case of a non-abelian group of order 6 they have size 1, 2 and 3, and there are correspondingly (for example) 50% of primes
1218:
921:
1271:
1904:
1348:
307:
1402:
382:
141:
68:
1869:
473:
393:
1375:
1245:
90:
in a general Galois extension is a major unsolved problem, the
Chebotarev density theorem says that the frequency of the occurrence of a given pattern, for all primes
1468:
1830:
1810:
1422:
1177:
2148:
825:
that have an order 2 element as their
Frobenius. So these primes have residue degree 2, so they split into exactly three prime ideals in a degree 6 extension of
1790:
1442:
329:
2153:
2430:
396:
demonstrates that this is indeed the case. Even though the prime numbers themselves appear rather erratically, splitting of the primes in the extension
2401:
Tschebotareff, N. (1926), "Die
Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören",
2346:
460:
established the framework for investigating this pattern and proved a special case of the theorem. The general statement was proved by
935:
192:
Then the theorem says that the asymptotic distribution of these invariants is uniform over the group, so that a conjugacy class with
2384:
575:
hence by the
Chebotarev density theorem, primes are asymptotically uniformly distributed among different residue classes coprime to
162:
among all primes. More generally, splitting behavior can be specified by assigning to (almost) every prime number an invariant, its
86:. There are only finitely many patterns of splitting that may occur. Although the full description of the splitting of every prime
2338:
461:
2435:
1118:{\displaystyle {\frac {|C|}{|G|}}{\Bigl (}\mathrm {li} (x)+O{\bigl (}{\sqrt {x}}(n\log x+\log |\Delta |){\bigr )}{\Bigr )},}
231:
is congruent to 1 mod 4, then it factors into a product of two distinct prime gaussian integers, or "splits completely"; if
926:
The statement is valid when the density refers to either the natural density or the analytic density of the set of primes.
402:
227:, he observed that the ordinary prime numbers may factor further in this new set of integers. In fact, if a prime
2029:
1276:
1914:
The statement of the
Chebotarev density theorem can be generalized to the case of an infinite Galois extension
456:. It turned out that the Galois group of the extension plays a key role in the pattern of splitting of primes.
1190:
667:
34:
2293:
521:
118:
886:
2094:
A consequence of this version of the theorem is that the
Frobenius elements of the unramified primes of
1250:
600:
1874:
730:
217:
2298:
2202:
1311:
253:
2005:
that is stable under conjugation and whose boundary has Haar measure zero. Then, the set of primes
1380:
706:
336:
103:
124:
51:
544:
2324:
1839:
1353:
1223:
2380:
2372:
2342:
784:
449:
163:
79:
1447:
2410:
2360:
2303:
2278:
1815:
596:
529:
453:
445:
221:
42:
2394:
2356:
2315:
1795:
1407:
1162:
17:
2390:
2364:
2352:
2311:
788:
765:
714:
619:
457:
167:
71:
1143:
The effective form of
Chebotarev's density theory becomes much weaker without GRH. Take
2123:
that split completely in it. A related corollary is that if almost all prime ideals of
1775:
1427:
314:
2226:
Lagarias, J.C.; Odlyzko, A.M. (1977). "Effective
Versions of the Chebotarev Theorem".
2424:
795:
675:
171:
75:
38:
2379:(Revised reprint of the 1968 original ed.), Wellesley, MA: A K Peters, Ltd.,
1833:
647:
27:
Describes statistically the splitting of primes in a given Galois extension of Q
1993:. The Chebotarev density theorem in this situation can be stated as follows:
1906:
term can be ignored. The implicit constant of this expression is absolute.
559:
is simply its residue class because the number of distinct primes into which
2171:
This particular example already follows from the
Frobenius result, because
543:
is abelian and can be canonically identified with the group of invertible
634:. Its 'splitting type' is the list of degrees of irreducible factors of
2414:
2307:
489:
235:
is congruent to 3 mod 4, then it remains prime, or is "inert"; and if
1220:
to be the Artin conductor of this representation. Suppose that, for
472:
The Chebotarev density theorem may be viewed as a generalisation of
524:. This is a special case of the Chebotarev density theorem for the
444:
Similar statistical laws also hold for splitting of primes in the
1970:
is compact in this topology, there is a unique Haar measure μ on
2091:
is finite (the Haar measure is then just the counting measure).
74:. Generally speaking, a prime integer will factor into several
1350:
is entire; that is, the Artin conjecture is satisfied for all
1128:
where the constant implied in the big-O notation is absolute,
2245:. Providence, RI: American Mathematical Society. p. 111.
1966:
is a profinite group equipped with the Krull topology. Since
476:. A quantitative form of Dirichlet's theorem states that if
591:
give an earlier result of Frobenius in this area. Suppose
239:
is 2 then it becomes a product of the square of the prime
805:
then the theorem says that asymptotically a proportion |
705:; in other words by choosing an ordering of α and its
2032:
1877:
1842:
1818:
1798:
1778:
1479:
1450:
1430:
1410:
1383:
1356:
1314:
1279:
1253:
1226:
1193:
1165:
979:
889:
405:
339:
317:
256:
127:
54:
857:
that is stable under conjugation. The set of primes
113:
A special case that is easier to state says that if
787:of a prime (ideal), which is in fact an associated
2377:Abelian l-adic representations and elliptic curves
2179:is more demanding than having the same cycle type.
2070:
1898:
1863:
1824:
1804:
1784:
1761:
1462:
1436:
1416:
1396:
1369:
1342:
1300:
1265:
1239:
1212:
1171:
1117:
915:
588:
430:
376:
323:
301:
147:, then the prime numbers that completely split in
135:
62:
1986:there is an associated Frobenius conjugacy class
1751:
1704:
1611:
1575:
1179:to be a nontrivial irreducible representation of
1107:
1016:
813:| of primes have associated Frobenius element as
752:states that for any given choice of Π the primes
431:{\displaystyle \mathbb {Z} \subset \mathbb {Z} }
2175:is a symmetric group. In general, conjugacy in
2119:is uniquely determined by the set of primes of
946:is a finite Galois extension with Galois group
869:and whose associated Frobenius conjugacy class
841:be a finite Galois extension of a number field
713:is faithfully represented as a subgroup of the
2149:Splitting of prime ideals in Galois extensions
934:The Generalized Riemann hypothesis implies an
474:Dirichlet's theorem on arithmetic progressions
394:Dirichlet's theorem on arithmetic progressions
196:elements occurs with frequency asymptotic to
166:, which is a representative of a well-defined
2334:Grundlehren der mathematischen Wissenschaften
1100:
1046:
107:
8:
2332:
2241:Iwaniec, Henryk; Kowalski, Emmanuel (2004).
2071:{\displaystyle {\frac {\mu (X)}{\mu (G)}}.}
1301:{\displaystyle \rho \otimes {\bar {\rho }}}
247:; we say that 2 "ramifies". For instance,
452:, with the Galois group isomorphic to the
2297:
2187:
2185:
2033:
2031:
1888:
1882:
1876:
1841:
1817:
1812:is trivial and is otherwise 0, and where
1797:
1777:
1750:
1749:
1731:
1730:
1703:
1702:
1686:
1668:
1667:
1638:
1616:
1610:
1609:
1586:
1580:
1574:
1573:
1540:
1535:
1525:
1504:
1503:
1484:
1478:
1449:
1429:
1409:
1388:
1382:
1361:
1355:
1325:
1313:
1287:
1286:
1278:
1252:
1231:
1225:
1195:
1194:
1192:
1164:
1106:
1105:
1099:
1098:
1090:
1082:
1051:
1045:
1044:
1021:
1015:
1014:
1006:
998:
991:
983:
980:
978:
890:
888:
415:
414:
407:
406:
404:
368:
338:
316:
255:
129:
128:
126:
56:
55:
53:
37:describes statistically the splitting of
2277:Lenstra, H. W.; Stevenhagen, P. (1996),
2154:Grothendieck–Katz p-curvature conjecture
1922:that is unramified outside a finite set
567:)/m, where m is multiplicative order of
2164:
614:) a monic integer polynomial such that
1213:{\displaystyle {\mathfrak {f}}(\rho )}
938:of the Chebotarev density theorem: if
2083:This reduces to the finite case when
1424:. Then there is an absolute positive
958:, the number of unramified primes of
768:δ, with δ equal to the proportion of
551:. The splitting invariant of a prime
106:in his thesis in 1922, published in (
7:
2279:"Chebotarëv and his density theorem"
646:factorizes in some fashion over the
496:, then the proportion of the primes
243:and the invertible gaussian integer
102:goes to infinity. It was proved by
2431:Theorems in algebraic number theory
1732:
1669:
1505:
1196:
1147:to be a finite Galois extension of
916:{\displaystyle {\frac {\#X}{\#G}}.}
1954:). In this case, the Galois group
1404:to be the character associated to
1266:{\displaystyle \rho \otimes \rho }
1087:
1025:
1022:
966:with Frobenius conjugacy class in
901:
893:
779:The statement of the more general
441:follows a simple statistical law.
25:
1899:{\displaystyle x^{\beta }/\beta }
697:is a permutation of the roots of
468:Relation with Dirichlet's theorem
2203:"The Chebotarev Density Theorem"
1871:; if there is no such zero, the
954:a union of conjugacy classes of
756:for which the splitting type of
589:Lenstra & Stevenhagen (1996)
462:Nikolai Grigoryevich Chebotaryov
2255:Corollary VII.13.10 of Neukirch
1946:is unramified in the extension
1930:(i.e. if there is a finite set
666:, then the splitting type is a
220:first introduced the notion of
121:which is a Galois extension of
2286:The Mathematical Intelligencer
2264:Corollary VII.13.7 of Neukirch
2059:
2053:
2045:
2039:
1858:
1846:
1746:
1743:
1737:
1724:
1709:
1680:
1674:
1635:
1625:
1546:
1531:
1516:
1510:
1343:{\displaystyle L(\rho _{0},s)}
1337:
1318:
1292:
1207:
1201:
1095:
1091:
1083:
1058:
1035:
1029:
1007:
999:
992:
984:
626:. It makes sense to factorise
535:. Indeed, the Galois group of
425:
419:
365:
352:
302:{\displaystyle 5=(1+2i)(1-2i)}
296:
281:
278:
263:
98:, tends to a certain limit as
1:
1397:{\displaystyle \chi _{\rho }}
733:, which gives a 'cycle type'
377:{\displaystyle 2=-i(1+i)^{2}}
136:{\displaystyle \mathbb {Q} }
63:{\displaystyle \mathbb {Q} }
31:Chebotarev's density theorem
18:Tchebotarev density theorem
2452:
2329:Algebraische Zahlentheorie
1864:{\displaystyle L(\rho ,s)}
1140:, and Δ its discriminant.
94:less than a large integer
2337:. Vol. 322. Berlin:
2201:Lenstra, Hendrik (2006).
1370:{\displaystyle \rho _{0}}
1240:{\displaystyle \rho _{0}}
829:with it as Galois group.
587:In their survey article,
776:that have cycle type Π.
741:), again a partition of
2228:Algebraic Number Fields
1938:such that any prime of
1463:{\displaystyle x\geq 2}
1247:a subrepresentation of
865:that are unramified in
674:. Considering also the
467:
35:algebraic number theory
2436:Analytic number theory
2333:
2243:Analytic Number Theory
2191:Section I.2.2 of Serre
2106:Important consequences
2072:
1900:
1865:
1826:
1825:{\displaystyle \beta }
1806:
1786:
1763:
1464:
1438:
1418:
1398:
1371:
1344:
1302:
1267:
1241:
1214:
1173:
1119:
917:
630:modulo a prime number
522:Euler totient function
432:
378:
325:
303:
212:History and motivation
137:
119:algebraic number field
64:
2403:Mathematische Annalen
2073:
1901:
1866:
1834:exceptional real zero
1827:
1807:
1805:{\displaystyle \rho }
1787:
1764:
1465:
1439:
1419:
1417:{\displaystyle \rho }
1399:
1372:
1345:
1303:
1268:
1242:
1215:
1174:
1172:{\displaystyle \rho }
1120:
918:
601:rational number field
446:cyclotomic extensions
433:
379:
326:
304:
138:
65:
2127:split completely in
2030:
1875:
1840:
1816:
1796:
1776:
1477:
1448:
1428:
1408:
1381:
1354:
1312:
1277:
1251:
1224:
1191:
1163:
977:
887:
750:theorem of Frobenius
731:cycle representation
707:algebraic conjugates
403:
337:
315:
254:
218:Carl Friedrich Gauss
125:
52:
1910:Infinite extensions
794:of elements of the
783:is in terms of the
508:is asymptotic to 1/
104:Nikolai Chebotaryov
2415:10.1007/BF01206606
2373:Serre, Jean-Pierre
2308:10.1007/BF03027290
2068:
1974:. For every prime
1896:
1861:
1822:
1802:
1782:
1759:
1520:
1460:
1434:
1414:
1394:
1367:
1340:
1298:
1263:
1237:
1210:
1169:
1151:with Galois group
1115:
913:
845:with Galois group
781:Chebotarev theorem
484:is an integer and
428:
374:
321:
309:splits completely;
299:
133:
108:Tschebotareff 1926
80:algebraic integers
60:
2348:978-3-540-65399-8
2063:
1785:{\displaystyle r}
1700:
1697:
1595:
1538:
1480:
1437:{\displaystyle c}
1295:
1132:is the degree of
1056:
1012:
936:effective version
930:Effective Version
908:
785:Frobenius element
662:is the degree of
324:{\displaystyle 3}
164:Frobenius element
16:(Redirected from
2443:
2417:
2397:
2368:
2336:
2325:Neukirch, Jürgen
2318:
2301:
2283:
2265:
2262:
2256:
2253:
2247:
2246:
2238:
2232:
2231:
2223:
2217:
2216:
2214:
2212:
2207:
2198:
2192:
2189:
2180:
2169:
2077:
2075:
2074:
2069:
2064:
2062:
2048:
2034:
2024:⊆ X has density
1905:
1903:
1902:
1897:
1892:
1887:
1886:
1870:
1868:
1867:
1862:
1831:
1829:
1828:
1823:
1811:
1809:
1808:
1803:
1791:
1789:
1788:
1783:
1768:
1766:
1765:
1760:
1755:
1754:
1736:
1735:
1708:
1707:
1701:
1699:
1698:
1687:
1673:
1672:
1656:
1646:
1645:
1617:
1615:
1614:
1596:
1591:
1590:
1581:
1579:
1578:
1545:
1544:
1539:
1536:
1530:
1529:
1519:
1509:
1508:
1469:
1467:
1466:
1461:
1443:
1441:
1440:
1435:
1423:
1421:
1420:
1415:
1403:
1401:
1400:
1395:
1393:
1392:
1376:
1374:
1373:
1368:
1366:
1365:
1349:
1347:
1346:
1341:
1330:
1329:
1307:
1305:
1304:
1299:
1297:
1296:
1288:
1272:
1270:
1269:
1264:
1246:
1244:
1243:
1238:
1236:
1235:
1219:
1217:
1216:
1211:
1200:
1199:
1178:
1176:
1175:
1170:
1124:
1122:
1121:
1116:
1111:
1110:
1104:
1103:
1094:
1086:
1057:
1052:
1050:
1049:
1028:
1020:
1019:
1013:
1011:
1010:
1002:
996:
995:
987:
981:
922:
920:
919:
914:
909:
907:
899:
891:
876:is contained in
729:by means of its
597:Galois extension
530:cyclotomic field
454:Klein four-group
437:
435:
434:
429:
418:
410:
383:
381:
380:
375:
373:
372:
330:
328:
327:
322:
308:
306:
305:
300:
222:complex integers
142:
140:
139:
134:
132:
72:rational numbers
69:
67:
66:
61:
59:
43:Galois extension
21:
2451:
2450:
2446:
2445:
2444:
2442:
2441:
2440:
2421:
2420:
2400:
2387:
2371:
2349:
2339:Springer-Verlag
2323:
2299:10.1.1.116.9409
2281:
2276:
2273:
2268:
2263:
2259:
2254:
2250:
2240:
2239:
2235:
2225:
2224:
2220:
2210:
2208:
2205:
2200:
2199:
2195:
2190:
2183:
2170:
2166:
2162:
2145:
2131:, then in fact
2108:
2049:
2035:
2028:
2027:
2023:
2001:be a subset of
1992:
1912:
1878:
1873:
1872:
1838:
1837:
1814:
1813:
1794:
1793:
1774:
1773:
1657:
1634:
1618:
1582:
1534:
1521:
1475:
1474:
1446:
1445:
1444:such that, for
1426:
1425:
1406:
1405:
1384:
1379:
1378:
1357:
1352:
1351:
1321:
1310:
1309:
1275:
1274:
1249:
1248:
1227:
1222:
1221:
1189:
1188:
1161:
1160:
997:
982:
975:
974:
932:
900:
892:
885:
884:
875:
853:be a subset of
835:
789:conjugacy class
766:natural density
725:. We can write
724:
715:symmetric group
657:
620:splitting field
585:
545:residue classes
470:
458:Georg Frobenius
401:
400:
364:
335:
334:
313:
312:
252:
251:
214:
168:conjugacy class
123:
122:
78:in the ring of
50:
49:
28:
23:
22:
15:
12:
11:
5:
2449:
2447:
2439:
2438:
2433:
2423:
2422:
2419:
2418:
2409:(1): 191–228,
2398:
2385:
2369:
2347:
2320:
2319:
2272:
2269:
2267:
2266:
2257:
2248:
2233:
2218:
2193:
2181:
2163:
2161:
2158:
2157:
2156:
2151:
2144:
2141:
2107:
2104:
2081:
2080:
2079:
2078:
2067:
2061:
2058:
2055:
2052:
2047:
2044:
2041:
2038:
2021:
1990:
1911:
1908:
1895:
1891:
1885:
1881:
1860:
1857:
1854:
1851:
1848:
1845:
1821:
1801:
1781:
1770:
1769:
1758:
1753:
1748:
1745:
1742:
1739:
1734:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1706:
1696:
1693:
1690:
1685:
1682:
1679:
1676:
1671:
1666:
1663:
1660:
1655:
1652:
1649:
1644:
1641:
1637:
1633:
1630:
1627:
1624:
1621:
1613:
1608:
1605:
1602:
1599:
1594:
1589:
1585:
1577:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1543:
1533:
1528:
1524:
1518:
1515:
1512:
1507:
1502:
1499:
1496:
1493:
1490:
1487:
1483:
1459:
1456:
1453:
1433:
1413:
1391:
1387:
1364:
1360:
1339:
1336:
1333:
1328:
1324:
1320:
1317:
1294:
1291:
1285:
1282:
1262:
1259:
1256:
1234:
1230:
1209:
1206:
1203:
1198:
1168:
1126:
1125:
1114:
1109:
1102:
1097:
1093:
1089:
1085:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1055:
1048:
1043:
1040:
1037:
1034:
1031:
1027:
1024:
1018:
1009:
1005:
1001:
994:
990:
986:
962:of norm below
931:
928:
924:
923:
912:
906:
903:
898:
895:
873:
834:
831:
720:
653:
584:
581:
469:
466:
439:
438:
427:
424:
421:
417:
413:
409:
386:
385:
371:
367:
363:
360:
357:
354:
351:
348:
345:
342:
332:
320:
310:
298:
295:
292:
289:
286:
283:
280:
277:
274:
271:
268:
265:
262:
259:
213:
210:
209:
208:
190:
189:
160:
159:
151:have density
131:
58:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2448:
2437:
2434:
2432:
2429:
2428:
2426:
2416:
2412:
2408:
2404:
2399:
2396:
2392:
2388:
2386:1-56881-077-6
2382:
2378:
2374:
2370:
2366:
2362:
2358:
2354:
2350:
2344:
2340:
2335:
2330:
2326:
2322:
2321:
2317:
2313:
2309:
2305:
2300:
2295:
2291:
2287:
2280:
2275:
2274:
2270:
2261:
2258:
2252:
2249:
2244:
2237:
2234:
2229:
2222:
2219:
2204:
2197:
2194:
2188:
2186:
2182:
2178:
2174:
2168:
2165:
2159:
2155:
2152:
2150:
2147:
2146:
2142:
2140:
2138:
2134:
2130:
2126:
2122:
2118:
2114:
2105:
2103:
2101:
2098:are dense in
2097:
2092:
2090:
2086:
2065:
2056:
2050:
2042:
2036:
2026:
2025:
2020:
2016:
2012:
2008:
2004:
2000:
1996:
1995:
1994:
1989:
1985:
1981:
1977:
1973:
1969:
1965:
1961:
1957:
1953:
1949:
1945:
1941:
1937:
1934:of primes of
1933:
1929:
1926:of primes of
1925:
1921:
1917:
1909:
1907:
1893:
1889:
1883:
1879:
1855:
1852:
1849:
1843:
1835:
1819:
1799:
1779:
1756:
1740:
1727:
1721:
1718:
1715:
1712:
1694:
1691:
1688:
1683:
1677:
1664:
1661:
1658:
1653:
1650:
1647:
1642:
1639:
1631:
1628:
1622:
1619:
1606:
1603:
1600:
1597:
1592:
1587:
1583:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1541:
1526:
1522:
1513:
1500:
1497:
1494:
1491:
1488:
1485:
1481:
1473:
1472:
1471:
1457:
1454:
1451:
1431:
1411:
1389:
1385:
1362:
1358:
1334:
1331:
1326:
1322:
1315:
1289:
1283:
1280:
1260:
1257:
1254:
1232:
1228:
1204:
1186:
1182:
1166:
1158:
1154:
1150:
1146:
1141:
1139:
1135:
1131:
1112:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1053:
1041:
1038:
1032:
1003:
988:
973:
972:
971:
969:
965:
961:
957:
953:
949:
945:
941:
937:
929:
927:
910:
904:
896:
883:
882:
881:
879:
872:
868:
864:
860:
856:
852:
848:
844:
840:
832:
830:
828:
824:
820:
816:
812:
808:
804:
800:
797:
793:
790:
786:
782:
777:
775:
771:
767:
763:
759:
755:
751:
746:
744:
740:
736:
732:
728:
723:
719:
716:
712:
708:
704:
700:
696:
692:
688:
684:
680:
677:
673:
669:
665:
661:
656:
652:
649:
645:
641:
637:
633:
629:
625:
621:
617:
613:
609:
605:
602:
598:
594:
590:
582:
580:
578:
574:
570:
566:
562:
558:
555:not dividing
554:
550:
546:
542:
538:
534:
531:
527:
523:
519:
515:
511:
507:
503:
500:congruent to
499:
495:
491:
487:
483:
479:
475:
465:
463:
459:
455:
451:
447:
442:
422:
411:
399:
398:
397:
395:
391:
369:
361:
358:
355:
349:
346:
343:
340:
333:
318:
311:
293:
290:
287:
284:
275:
272:
269:
266:
260:
257:
250:
249:
248:
246:
242:
238:
234:
230:
226:
223:
219:
211:
206:
202:
199:
198:
197:
195:
187:
183:
179:
176:
175:
174:
173:
169:
165:
158:
154:
153:
152:
150:
146:
120:
116:
111:
109:
105:
101:
97:
93:
89:
85:
81:
77:
73:
48:of the field
47:
44:
40:
36:
32:
19:
2406:
2402:
2376:
2328:
2292:(2): 26–37,
2289:
2285:
2260:
2251:
2242:
2236:
2227:
2221:
2209:. Retrieved
2196:
2176:
2172:
2167:
2136:
2132:
2128:
2124:
2120:
2116:
2112:
2109:
2099:
2095:
2093:
2088:
2084:
2082:
2018:
2014:
2010:
2006:
2002:
1998:
1987:
1983:
1979:
1975:
1971:
1967:
1963:
1959:
1955:
1951:
1947:
1943:
1939:
1935:
1931:
1927:
1923:
1919:
1915:
1913:
1771:
1184:
1180:
1156:
1152:
1148:
1144:
1142:
1137:
1133:
1129:
1127:
967:
963:
959:
955:
951:
947:
943:
939:
933:
925:
880:has density
877:
870:
866:
862:
858:
854:
850:
846:
842:
838:
836:
826:
822:
818:
814:
810:
806:
802:
801:. If we fix
798:
796:Galois group
791:
780:
778:
773:
769:
761:
757:
753:
749:
747:
742:
738:
734:
726:
721:
717:
710:
702:
698:
694:
690:
686:
682:
678:
676:Galois group
671:
663:
659:
654:
650:
643:
639:
635:
631:
627:
623:
615:
611:
607:
603:
592:
586:
576:
572:
568:
564:
563:splits is φ(
560:
556:
552:
548:
540:
536:
532:
525:
517:
513:
509:
505:
501:
497:
493:
485:
481:
477:
471:
443:
440:
389:
387:
244:
240:
236:
232:
228:
224:
215:
204:
200:
193:
191:
185:
181:
177:
172:Galois group
161:
156:
148:
144:
114:
112:
99:
95:
91:
87:
83:
76:ideal primes
45:
30:
29:
1187:, and take
1155:and degree
764:is Π has a
648:prime field
583:Formulation
41:in a given
2425:Categories
2365:0956.11021
2271:References
2230:: 409–464.
2017:such that
1183:of degree
143:of degree
2375:(1998) ,
2294:CiteSeerX
2051:μ
2037:μ
1894:β
1884:β
1850:ρ
1820:β
1800:ρ
1741:ρ
1722:
1692:
1678:ρ
1665:
1651:
1640:−
1620:−
1607:
1593:β
1588:β
1553:
1527:ρ
1523:χ
1514:ρ
1489:≤
1482:∑
1455:≥
1412:ρ
1390:ρ
1386:χ
1359:ρ
1323:ρ
1293:¯
1290:ρ
1284:⊗
1281:ρ
1261:ρ
1258:⊗
1255:ρ
1229:ρ
1205:ρ
1167:ρ
1088:Δ
1080:
1068:
902:#
894:#
833:Statement
668:partition
520:) is the
464:in 1922.
412:⊂
384:ramifies.
347:−
331:is inert;
288:−
2327:(1999).
2143:See also
1792:is 1 if
1501:∤
512:, where
2395:1484415
2357:1697859
2316:1395088
2013:not in
1982:not in
1942:not in
1377:. Take
1159:. Take
817:. When
689:, each
642:, i.e.
599:of the
571:modulo
490:coprime
450:abelian
170:in the
2393:
2383:
2363:
2355:
2345:
2314:
2296:
2211:7 June
1832:is an
1772:where
950:, and
849:. Let
606:, and
117:is an
39:primes
2282:(PDF)
2206:(PDF)
2160:Notes
1136:over
685:over
670:Π of
658:. If
618:is a
595:is a
241:(1+i)
216:When
2381:ISBN
2343:ISBN
2213:2018
1997:Let
837:Let
760:mod
748:The
638:mod
547:mod
504:mod
2411:doi
2361:Zbl
2304:doi
2009:of
1978:of
1958:of
1836:of
1719:log
1689:log
1662:log
1648:log
1604:exp
1550:log
1273:or
1077:log
1065:log
970:is
861:of
809:|/|
772:in
701:in
693:in
681:of
622:of
528:th
516:=φ(
492:to
488:is
178:Gal
110:).
82:of
70:of
33:in
2427::
2407:95
2405:,
2391:MR
2389:,
2359:.
2353:MR
2351:.
2341:.
2331:.
2312:MR
2310:,
2302:,
2290:18
2288:,
2284:,
2184:^
2139:.
2135:=
2115:,
2102:.
2087:/
1962:/
1950:/
1918:/
1537:Fr
1470:,
1308:,
745:.
709:,
579:.
573:N;
392:.
245:-i
188:).
155:1/
2413::
2367:.
2306::
2215:.
2177:G
2173:G
2137:K
2133:L
2129:L
2125:K
2121:K
2117:L
2113:K
2100:G
2096:L
2089:K
2085:L
2066:.
2060:)
2057:G
2054:(
2046:)
2043:X
2040:(
2022:v
2019:F
2015:S
2011:K
2007:v
2003:G
1999:X
1991:v
1988:F
1984:S
1980:K
1976:v
1972:G
1968:G
1964:K
1960:L
1956:G
1952:K
1948:L
1944:S
1940:K
1936:K
1932:S
1928:K
1924:S
1920:K
1916:L
1890:/
1880:x
1859:)
1856:s
1853:,
1847:(
1844:L
1780:r
1757:,
1752:)
1747:)
1744:)
1738:(
1733:f
1728:x
1725:(
1716:n
1713:d
1710:(
1705:)
1695:x
1684:+
1681:)
1675:(
1670:f
1659:3
1654:x
1643:4
1636:)
1632:n
1629:d
1626:(
1623:c
1612:(
1601:x
1598:+
1584:x
1576:(
1571:O
1568:+
1565:x
1562:r
1559:=
1556:p
1547:)
1542:p
1532:(
1517:)
1511:(
1506:f
1498:p
1495:,
1492:x
1486:p
1458:2
1452:x
1432:c
1363:0
1338:)
1335:s
1332:,
1327:0
1319:(
1316:L
1233:0
1208:)
1202:(
1197:f
1185:n
1181:G
1157:d
1153:G
1149:Q
1145:L
1138:Q
1134:L
1130:n
1113:,
1108:)
1101:)
1096:)
1092:|
1084:|
1074:+
1071:x
1062:n
1059:(
1054:x
1047:(
1042:O
1039:+
1036:)
1033:x
1030:(
1026:i
1023:l
1017:(
1008:|
1004:G
1000:|
993:|
989:C
985:|
968:C
964:x
960:K
956:G
952:C
948:G
944:K
942:/
940:L
911:.
905:G
897:X
878:X
874:v
871:F
867:L
863:K
859:v
855:G
851:X
847:G
843:K
839:L
827:Q
823:p
819:G
815:C
811:G
807:C
803:C
799:G
792:C
774:G
770:g
762:p
758:P
754:p
743:n
739:g
737:(
735:c
727:g
722:n
718:S
711:G
703:K
699:P
695:G
691:g
687:Q
683:K
679:G
672:n
664:P
660:n
655:p
651:F
644:P
640:p
636:P
632:p
628:P
624:P
616:K
612:t
610:(
608:P
604:Q
593:K
577:N
569:p
565:N
561:p
557:N
553:p
549:N
541:Q
539:/
537:K
533:K
526:N
518:N
514:n
510:n
506:N
502:a
498:p
494:N
486:a
482:2
480:≥
478:N
426:]
423:i
420:[
416:Z
408:Z
390:Z
370:2
366:)
362:i
359:+
356:1
353:(
350:i
344:=
341:2
319:3
297:)
294:i
291:2
285:1
282:(
279:)
276:i
273:2
270:+
267:1
264:(
261:=
258:5
237:p
233:p
229:p
225:Z
207:.
205:n
203:/
201:k
194:k
186:Q
184:/
182:K
180:(
157:n
149:K
145:n
130:Q
115:K
100:N
96:N
92:p
88:p
84:K
57:Q
46:K
20:)
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