954:
of a number field is the group of fractional ideals in the ring of integers in the field modulo principal ideals. The cardinality of the group is called the class number of the number field. It measures the extent of the failure of unique
1140:
is an extension of the class field theory (which is about abelian extensions of number fields) to non-abelian extensions; or at least the idea of such a theory. The non-abelian theory does not exist in a definitive form
1230:
1252:
998:
501:
33:
1454:
is a prime number that is 2 less or 2 more than another prime number. For example, 7 is a twin prime, since it is prime and 5 is also prime.
922:
432:
1554:
864:
1204:
1137:
974:
2. In algebraic number theory, an integer sometimes means an element of a ring of integers; e.g., a
450:
273:
1128:, also called an algebraic number field, is a finite-degree field extension of the field of rational numbers.
1039:
of a finite list of integers is the smallest positive number that is a multiple of every integer in the list.
813:
is the process of splitting a mathematical object, often integers or polynomials, into a product of factors.
907:
of a finite list of integers is the largest positive number that is a divisor of every integer in the list.
867:
states that every integer greater than 1 can be written uniquely (up to reordering) as a product of primes.
538:
418:
40:
1405:
is an integer that is the square of an integer. For example, 4 and 9 are squares, but 10 is not a square.
1170:
is an equivalence class of non-Archimedean valuations (finite place) or absolute values (infinite place).
723:
525:
513:
425:
581:. Divisors can be defined in exactly the same way for polynomials or for elements of a commutative ring.
1065:
853:
824:, one of the most famous and difficult to prove theorems in number theory, states that for any integer
395:
1431:
895:
is a conjecture that states that every even natural number greater than 2 is the sum of two primes.
892:
821:
1235:
981:
616:
904:
252:
307:
1549:
1413:
1125:
479:
262:
1491:
1379:
1352:
1167:
1036:
277:
1186:
667:
490:
439:
378:
29:
8:
1508:
1469:
1390:
1310:
507:
25:
464:(also called relatively prime) if the only positive integer that divides them both is 1.
1263:
1156:
1106:
677:
435:
is the number of equivalence classes of binary quadratic forms of a given discriminant.
406:
1362:
1198:
1025:
951:
626:
596:
1334:
1330:
975:
289:
234:
1047:
1084:
1008:
606:
214:
1543:
1402:
933:
810:
17:
1480:
1179:
1095:
882:
421:
of a number field is the cardinality of the ideal class group of the field.
1058:
21:
1451:
1061:
in number theory is the completion of a number field at a finite place.
224:
46:
964:
550:
461:
1393:
is an integer that is not divisible by any square other than 1.
1182:
is a positive integer with no divisors other than itself and 1.
541:
is a generalization/variant of a distribution in analysis.
1416:
is, in a modified form, equivalent to the abc conjecture.
1109:(which used to be called the Taniyama–Shimura conjecture)
1189:
describes the asymptotic distribution of prime numbers.
708:. Euler's theorem generalizes Fermat's little theorem.
1238:
1207:
984:
857:
280:
is entire (holomorphic on the entire complex plane).
1246:
1224:
992:
1541:
1333:in a number field is the ring consisting of all
710:
316:is the greatest common divisor of two integers
34:Glossary of arithmetic and diophantine geometry
978:. To avoid ambiguity, an integer contained in
847:
518:
502:Dirichlet's theorem on arithmetic progressions
428:is the number of conjugacy classes of a group.
16:This is a glossary of concepts and results in
1087:is a prime number one less than a power of 2.
409:concerns abelian extensions of number fields.
310:, also called BĂ©zout's lemma, states that if
886:
815:
389:
1201:is an element in the profinite completion
897:
246:
1240:
1225:{\displaystyle {\widehat {\mathbb {Z} }}}
1212:
986:
256:
238:
1373:
1346:
1029:
661:
301:
1502:
1000:is sometimes called a rational integer.
743:, is the number of integers coprime to
484:
354:, and in fact the integers of the form
1542:
1383:
1150:
671:
1256:
1099:
620:
1356:
692:are coprime positive integers, then
635:divides the product of two integers
1436:
1434:is a theorem in class field theory.
1323:
1315:
424:2. In group theory, the
282:
228:
13:
1483:is a field with a valuation on it.
1041:
923:Hasse's theorem on elliptic curves
845:has no positive integer solutions.
565:such that there exists an integer
529:is a book by Carl Friedrich Gauss.
292:is a certain holomorphic function.
14:
1566:
1077:
865:fundamental theorem of arithmetic
859:fundamental theorem of arithmetic
600:
1395:
804:
1473:
1303:
1172:
1118:
1089:
876:
531:
411:
49:
1138:non-abelian class field theory
1130:
451:Conductor (class field theory)
399:
1:
1444:
1442:See Euler's totient function.
590:
539:distribution in number theory
364:are exactly the multiples of
328:, then there exists integers
1463:
1247:{\displaystyle \mathbb {Z} }
1191:
1019:
993:{\displaystyle \mathbb {Q} }
969:…, -3, -2, -1, 0, 1, 2, 3, …
647:must divide at least one of
494:
444:
48:
41:List of number theory topics
7:
1266:is three positive integers
526:Disquisitiones Arithmeticae
520:Disquisitiones Arithmeticae
473:
10:
1571:
1002:
957:
802:See the entry for divisor.
543:
454:
372:
24:. Concepts and results in
1555:Glossaries of mathematics
1425:
1407:
796:
396:Chinese remainder theorem
391:Chinese remainder theorem
1518:is prime if and only if
1485:
1432:Takagi existence theorem
1160:
1051:
944:
927:
916:
759:inclusive. For example,
724:Euler's totient function
712:Euler's totient function
610:
553:or factor of an integer
514:Dirichlet's unit theorem
266:
218:
61:
56:
1337:contained in the field.
905:greatest common divisor
899:greatest common divisor
854:Fermat's little theorem
849:Fermat's little theorem
716:For a positive integer
629:states that if a prime
253:Algebraic number theory
248:algebraic number theory
1248:
1226:
1066:local–global principle
994:
480:Dedekind zeta function
263:Analytic number theory
258:analytic number theory
240:algebraic number field
208:
191:
186:
181:
176:
171:
166:
161:
156:
151:
146:
141:
136:
131:
126:
121:
116:
111:
106:
101:
96:
91:
86:
81:
76:
71:
66:
1380:Sieve of Eratosthenes
1375:sieve of Eratosthenes
1353:Quadratic reciprocity
1348:quadratic reciprocity
1249:
1227:
1037:least common multiple
1031:least common multiple
995:
893:Goldbach's conjecture
888:Goldbach's conjecture
822:Fermat's last theorem
817:Fermat's last theorem
1497:
1458:
1420:
1368:
1341:
1298:
1236:
1205:
1187:prime number theorem
1145:
1113:
1072:
1014:
982:
939:
911:
871:
791:
585:
491:Diophantine equation
486:Diophantine equation
468:
440:class number problem
384:
296:
203:
30:diophantine geometry
1470:valuation (algebra)
1414:Szpiro's conjecture
1391:square-free integer
1385:square-free integer
1311:ramification theory
1254:along all integers.
508:Dirichlet character
50:Contents:
26:arithmetic geometry
1492:Vojta's conjecture
1264:Pythagorean triple
1258:Pythagorean triple
1244:
1222:
1185:2. The
1107:modularity theorem
1101:modularity theorem
1064:2. The
990:
963:1. The
438:4. The
417:1. The
407:class field theory
278:Artin's L function
1363:Quadratic residue
1358:quadratic residue
1335:algebraic numbers
1219:
1199:profinite integer
1026:Langlands program
952:ideal class group
668:Euler's criterion
663:Euler's criterion
617:Erdős–Kac theorem
597:Eisenstein series
460:Two integers are
379:Brocard's problem
308:BĂ©zout's identity
303:BĂ©zout's identity
244:See number field.
1562:
1535:
1529:
1526:is congruent to
1525:
1517:
1509:Wilson's theorem
1504:Wilson's theorem
1438:totient function
1331:ring of integers
1325:ring of integers
1317:relatively prime
1293:
1279:
1253:
1251:
1250:
1245:
1243:
1231:
1229:
1228:
1223:
1221:
1220:
1215:
1210:
1178:1. A
1057:1. A
999:
997:
996:
991:
989:
976:Gaussian integer
970:
967:are the numbers
844:
830:
786:
780:
765:
758:
752:
748:
742:
731:
721:
707:
701:
698:is congruent to
697:
691:
685:
658:
652:
646:
640:
634:
580:
570:
564:
558:
369:
363:
353:
339:
333:
327:
321:
315:
290:automorphic form
284:automorphic form
274:Artin conjecture
235:Algebraic number
230:algebraic number
51:
32:can be found in
1570:
1569:
1565:
1564:
1563:
1561:
1560:
1559:
1540:
1539:
1531:
1527:
1519:
1512:
1505:
1500:
1488:
1476:
1466:
1461:
1447:
1439:
1428:
1423:
1410:
1398:
1386:
1376:
1371:
1359:
1349:
1344:
1326:
1318:
1306:
1301:
1281:
1267:
1259:
1239:
1237:
1234:
1233:
1211:
1209:
1208:
1206:
1203:
1202:
1194:
1175:
1163:
1157:Pell's equation
1153:
1152:Pell's equation
1148:
1133:
1121:
1116:
1102:
1092:
1080:
1075:
1054:
1048:Legendre symbol
1044:
1043:Legendre symbol
1032:
1022:
1017:
1005:
985:
983:
980:
979:
968:
960:
947:
942:
930:
919:
914:
900:
889:
879:
874:
860:
850:
832:
831:, the equation
825:
818:
807:
799:
794:
782:
767:
760:
754:
750:
744:
733:
727:
717:
713:
703:
699:
693:
687:
681:
680:states that if
678:Euler's theorem
674:
673:Euler's theorem
664:
654:
648:
642:
636:
630:
623:
613:
603:
593:
588:
572:
566:
560:
554:
546:
534:
521:
497:
487:
476:
471:
457:
447:
414:
402:
392:
387:
375:
365:
355:
341:
335:
329:
323:
317:
311:
304:
299:
285:
269:
259:
249:
241:
231:
221:
211:
206:
201:
200:
199:
198:
52:
12:
11:
5:
1568:
1558:
1557:
1552:
1538:
1537:
1506:
1503:
1499:
1496:
1495:
1494:
1489:
1486:
1484:
1477:
1474:
1472:
1467:
1464:
1460:
1457:
1456:
1455:
1448:
1445:
1443:
1440:
1437:
1435:
1429:
1426:
1422:
1419:
1418:
1417:
1411:
1408:
1406:
1399:
1396:
1394:
1387:
1384:
1382:
1377:
1374:
1370:
1367:
1366:
1365:
1360:
1357:
1355:
1350:
1347:
1343:
1340:
1339:
1338:
1327:
1324:
1322:
1319:
1316:
1314:
1307:
1304:
1300:
1297:
1296:
1295:
1260:
1257:
1255:
1242:
1218:
1214:
1195:
1192:
1190:
1183:
1176:
1173:
1171:
1164:
1161:
1159:
1154:
1151:
1147:
1144:
1143:
1142:
1134:
1131:
1129:
1122:
1119:
1115:
1112:
1111:
1110:
1103:
1100:
1098:
1093:
1090:
1088:
1085:Mersenne prime
1081:
1079:Mersenne prime
1078:
1074:
1071:
1070:
1069:
1062:
1055:
1052:
1050:
1045:
1042:
1040:
1033:
1030:
1028:
1023:
1020:
1016:
1013:
1012:
1011:
1009:Iwasawa theory
1006:
1003:
1001:
988:
972:
961:
958:
956:
955:factorization.
948:
945:
941:
938:
937:
936:
931:
928:
926:
920:
917:
913:
910:
909:
908:
901:
898:
896:
890:
887:
885:
880:
877:
873:
870:
869:
868:
861:
858:
856:
851:
848:
846:
819:
816:
814:
808:
805:
803:
800:
797:
793:
790:
789:
788:
781:for any prime
714:
711:
709:
675:
672:
670:
665:
662:
660:
627:Euclid's lemma
624:
622:Euclid's lemma
621:
619:
614:
611:
609:
607:Elliptic curve
604:
602:elliptic curve
601:
599:
594:
591:
587:
584:
583:
582:
559:is an integer
547:
544:
542:
535:
532:
530:
522:
519:
517:
512:3.
510:
506:2.
504:
500:1.
498:
495:
493:
488:
485:
483:
477:
474:
470:
467:
466:
465:
458:
455:
453:
448:
445:
443:
436:
431:3.
429:
422:
415:
412:
410:
403:
400:
398:
393:
390:
386:
383:
382:
381:
376:
373:
371:
305:
302:
298:
295:
294:
293:
286:
283:
281:
270:
267:
265:
260:
257:
255:
250:
247:
245:
242:
239:
237:
232:
229:
227:
222:
219:
217:
215:abc conjecture
212:
209:
205:
202:
195:
194:
189:
184:
179:
174:
169:
164:
159:
154:
149:
144:
139:
134:
129:
124:
119:
114:
109:
104:
99:
94:
89:
84:
79:
74:
69:
64:
59:
53:
47:
45:
9:
6:
4:
3:
2:
1567:
1556:
1553:
1551:
1550:Number theory
1548:
1547:
1545:
1534:
1523:
1515:
1510:
1507:
1501:
1493:
1490:
1482:
1478:
1471:
1468:
1462:
1453:
1449:
1441:
1433:
1430:
1424:
1415:
1412:
1404:
1403:square number
1400:
1397:square number
1392:
1388:
1381:
1378:
1372:
1364:
1361:
1354:
1351:
1345:
1336:
1332:
1328:
1320:
1312:
1308:
1302:
1292:
1288:
1284:
1278:
1274:
1270:
1265:
1261:
1216:
1200:
1196:
1188:
1184:
1181:
1177:
1169:
1165:
1158:
1155:
1149:
1139:
1135:
1127:
1123:
1117:
1108:
1104:
1097:
1094:
1086:
1082:
1076:
1067:
1063:
1060:
1056:
1049:
1046:
1038:
1034:
1027:
1024:
1018:
1010:
1007:
977:
973:
966:
962:
953:
949:
943:
935:
932:
924:
921:
915:
906:
902:
894:
891:
884:
881:
875:
866:
862:
855:
852:
843:
839:
835:
828:
823:
820:
812:
811:Factorization
809:
806:factorization
801:
795:
785:
778:
774:
770:
763:
757:
747:
740:
736:
730:
725:
720:
715:
706:
696:
690:
684:
679:
676:
669:
666:
657:
651:
645:
639:
633:
628:
625:
618:
615:
608:
605:
598:
595:
589:
579:
575:
569:
563:
557:
552:
548:
540:
536:
528:
527:
523:
515:
511:
509:
505:
503:
499:
492:
489:
481:
478:
472:
463:
459:
452:
449:
441:
437:
434:
430:
427:
423:
420:
416:
408:
404:
397:
394:
388:
380:
377:
368:
362:
358:
352:
348:
344:
338:
332:
326:
320:
314:
309:
306:
300:
291:
287:
279:
275:
271:
264:
261:
254:
251:
243:
236:
233:
226:
223:
216:
213:
207:
197:
193:
190:
188:
185:
183:
180:
178:
175:
173:
170:
168:
165:
163:
160:
158:
155:
153:
150:
148:
145:
143:
140:
138:
135:
133:
130:
128:
125:
123:
120:
118:
115:
113:
110:
108:
105:
103:
100:
98:
95:
93:
90:
88:
85:
83:
80:
78:
75:
73:
70:
68:
65:
63:
60:
58:
55:
54:
44:
42:
37:
35:
31:
27:
23:
20:, a field of
19:
18:number theory
1532:
1521:
1513:
1511:states that
1481:valued field
1475:valued field
1321:See coprime.
1305:ramification
1290:
1286:
1282:
1276:
1272:
1268:
1180:prime number
1174:prime number
1126:number field
1120:number field
1096:Modular form
1091:modular form
883:Global field
878:global field
841:
837:
833:
826:
783:
776:
772:
768:
761:
755:
745:
738:
734:
728:
718:
704:
694:
688:
682:
655:
649:
643:
637:
631:
577:
573:
567:
561:
555:
533:distribution
524:
433:Class number
426:class number
419:class number
413:class number
366:
360:
356:
350:
346:
342:
336:
330:
324:
318:
312:
196:
38:
15:
1132:non-abelian
1059:local field
571:satisfying
401:class field
22:mathematics
1544:Categories
1452:twin prime
1446:twin prime
1280:such that
934:Hecke ring
732:, denoted
592:Eisenstein
340:such that
225:Adele ring
1465:valuation
1217:^
1193:profinite
1021:Langlands
496:Dirichlet
446:conductor
39:See also
965:integers
749:between
475:Dedekind
1004:Iwasawa
959:integer
764:(4) = 2
641:, then
551:divisor
545:divisor
462:coprime
456:coprime
374:Brocard
1516:> 1
1427:Takagi
1409:Szpiro
1141:today.
829:> 2
798:factor
1487:Vojta
1168:place
1162:place
1053:local
946:ideal
929:Hecke
918:Hasse
612:Erdős
276:says
268:Artin
220:adele
1530:mod
1524:-1)!
1329:The
1309:The
1136:The
1105:The
1035:The
950:The
903:The
863:The
775:) =
766:and
753:and
702:mod
686:and
405:The
334:and
322:and
272:The
28:and
1232:of
779:- 1
726:of
653:or
288:An
210:abc
62:0–9
57:Top
1546::
1528:-1
1479:A
1450:A
1401:A
1389:A
1289:=
1285:+
1275:,
1271:,
1262:A
1197:A
1166:A
1124:A
1083:A
840:=
836:+
722:,
638:ab
578:mk
576:=
549:A
537:A
361:bt
359:+
357:as
349:=
347:by
345:+
343:ax
43:.
36:.
1536:.
1533:n
1522:n
1520:(
1514:n
1498:W
1459:V
1421:T
1369:S
1342:Q
1313:.
1299:R
1294:.
1291:c
1287:b
1283:a
1277:c
1273:b
1269:a
1241:Z
1213:Z
1146:P
1114:N
1073:M
1068:.
1015:L
987:Q
971:.
940:I
925:.
912:H
872:G
842:c
838:b
834:a
827:n
792:F
787:.
784:p
777:p
773:p
771:(
769:φ
762:φ
756:n
751:1
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739:n
737:(
735:φ
729:n
719:n
705:n
700:1
695:a
689:a
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659:.
656:b
650:a
644:p
632:p
586:E
574:n
568:k
562:m
556:n
516:.
482:.
469:D
442:.
385:C
370:.
367:d
351:d
337:y
331:x
325:b
319:a
313:d
297:B
204:A
192:Z
187:Y
182:X
177:W
172:V
167:U
162:T
157:S
152:R
147:Q
142:P
137:O
132:N
127:M
122:L
117:K
112:J
107:I
102:H
97:G
92:F
87:E
82:D
77:C
72:B
67:A
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