1813:
1586:
1985:
943:
1808:{\displaystyle {\begin{cases}\left({\frac {L/K}{\cdot }}\right):I_{K}^{\Delta }\longrightarrow \operatorname {Gal} (L/K)\\\prod _{i=1}^{m}{\mathfrak {p}}_{i}^{n_{i}}\longmapsto \prod _{i=1}^{m}\left({\frac {L/K}{{\mathfrak {p}}_{i}}}\right)^{n_{i}}\end{cases}}}
3468:
552:
284:
3838:
3066:
1843:
4082:
790:
1415:
1362:
4119:
Frei, Günther (2004), "On the history of the Artin reciprocity law in abelian extensions of algebraic number fields: how Artin was led to his reciprocity law", in Olav
Arnfinn Laudal; Ragni Piene (eds.),
2797:
4383:
Algebraic number theory. Proceedings of an instructional conference organized by the London
Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union
4356:
Algebraic number theory. Proceedings of an instructional conference organized by the London
Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union
3346:
2377:
3291:
1510:
2997:
3232:
2901:
2723:
3527:
805:
2675:
2291:
3687:
3117:
2491:
2534:
2415:
1463:
3161:
2249:
2625:
353:
3358:
2135:
2057:
3843:
where the left hand side is the Artin L-function associated to the extension with character σ and the right hand side is the Hecke L-function associated with χ, Section 7.D of.
1309:
2208:
1578:
1272:
1236:
1212:
1180:
2830:
2449:
982:
2317:
1145:
692:
645:
309:
2573:
1101:
2173:
1006:
618:
417:
3715:
2932:
438:
140:
3636:
105:
665:
598:
578:
397:
377:
164:
179:
3727:
3005:
1980:{\displaystyle I_{K}^{\mathbf {c} }/i(K_{\mathbf {c} ,1})\mathrm {N} _{L/K}(I_{L}^{\mathbf {c} }){\overset {\sim }{\longrightarrow }}\mathrm {Gal} (L/K)}
1069:
in them. Therefore, the Artin reciprocity law can be interpreted as one of the main theorems of global class field theory. It can be used to prove that
1092:
for ideal classes of algebraic number fields into the group theoretic task of determining the kernels of transfers of finite non-abelian groups.
716:
4229:
4196:
4133:
1367:
1314:
1102:
https://math.stackexchange.com/questions/4131855/frobenius-elements#:~:text=A%20Frobenius%20element%20for%20P,some%20%CF%84%E2%88%88KP
4326:
4287:
4167:
1827:
2738:
4122:
The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3--8, 2002
3300:
2341:
3237:
1468:
2937:
938:{\displaystyle {\hat {H}}^{0}(\operatorname {Gal} (L/K),C_{L})\simeq {\hat {H}}^{-2}(\operatorname {Gal} (L/K),\mathbb {Z} ),}
3186:
2855:
2083:
2680:
2451:
is the principal ideal (Δ) or (Δ)∞ according to whether Δ is positive or negative, and the Artin map on a prime-to-Δ ideal (
4278:, Grundlehren der Mathematischen Wissenschaften, vol. 322, Translated from the German by Norbert Schappacher, Berlin:
3484:
4184:
2638:
2254:
1085:
44:
3651:
3074:
2458:
2504:
2385:
4406:
1428:
3122:
1545:
1089:
1078:
68:
4370:
3905:, Springer, 1992, p. 408. In fact, a more precise version of the reciprocity law keeps track of the ramification.
3593:
3463:{\displaystyle \left({\frac {F/\mathbb {Q} }{(n)}}\right)=\left({\frac {L/\mathbb {Q} }{(n)}}\right){\pmod {H}}.}
2213:
48:
2586:
1050:
699:
558:
318:
3646:
2576:
2096:
2028:
1021:
143:
1513:
1017:
985:
4213:
3071:
The relation between the quadratic and Artin reciprocity laws is given by studying the quadratic field
1285:
4378:
4351:
3690:
2182:
1551:
1253:
1217:
1193:
1161:
2805:
2424:
1595:
951:
47:" refers to a long line of more concrete number theoretic statements which it generalized, from the
1187:
1066:
1027:
429:
52:
2300:
1128:
2176:
670:
623:
292:
40:
4271:
3898:
2551:
1084:
Two years after the publication of his general reciprocity law in 1927, Artin rediscovered the
710:
A cohomological proof of the global reciprocity law can be achieved by first establishing that
4343:
4322:
4304:
4283:
4225:
4192:
4163:
4129:
3554:
1152:
1111:
1054:
1046:
547:{\displaystyle \theta _{v}:K_{v}^{\times }/N_{L_{v}/K_{v}}(L_{v}^{\times })\to G^{\text{ab}},}
2152:
991:
603:
402:
4386:
4374:
4359:
4347:
4332:
4314:
4293:
4243:
4147:
4060:, Annals of Mathematics Studies, vol. 83, Princeton, N.J.: Princeton University Press,
3967:
3589:
3585:
3558:
2632:
2418:
1108:
1070:
108:
4239:
4206:
4143:
4065:
3700:
2917:
118:
4390:
4363:
4336:
4318:
4297:
4279:
4247:
4235:
4221:
4202:
4188:
4151:
4139:
4125:
4061:
4053:
3569:
2001:
1042:
796:
3613:
82:
279:{\displaystyle \theta :C_{K}/{N_{L/K}(C_{L})}\to \operatorname {Gal} (L/K)^{\text{ab}},}
4107:
Emil Artin (1930) "Idealklassen in Oberkörpern und allgemeines
Reziprozitätsgesetzes",
4090:
3833:{\displaystyle L_{E/K}^{\mathrm {Artin} }(\sigma ,s)=L_{K}^{\mathrm {Hecke} }(\chi ,s)}
3577:
650:
583:
563:
382:
362:
312:
149:
64:
4400:
1247:
60:
36:
17:
3958:(December 1929), "Idealklassen in oberkörpern und allgemeines reziprozitätsgesetz",
3061:{\displaystyle \left({\frac {\ell ^{*}}{p}}\right)=\left({\frac {p}{\ell }}\right).}
4309:
3869:
3639:
3562:
1122:
1035:
1031:
356:
112:
56:
4036:
1311:
is canonically isomorphic to the Galois group of the extension of residue fields
1148:
1074:
3854:-dimensional representations, though a direct correspondence is still lacking.
4255:
4176:
3955:
3600:
1275:
32:
3348:
A basic property of the Artin symbol says that for every prime-to-ℓ ideal (
3565:
with Hecke L-functions associated to characters of the idèle class group.
785:{\displaystyle (\operatorname {Gal} (K^{\text{sep}}/K),\varinjlim C_{L})}
3880:, edited by Cassels and Frölich, Academic Press, 1967, pp. 266–279
3971:
2999:(which is always 1 (mod 4)). Then, quadratic reciprocity states that
1417:. There is therefore a canonically defined Frobenius element in Gal(
4109:
4098:
4083:
3960:
4096:
Emil Artin (1927) "Beweis des allgemeinen
Reziprozitätsgesetzes",
1410:{\displaystyle {\mathcal {O}}_{K,{\mathfrak {p}}}/{\mathfrak {p}}}
1357:{\displaystyle {\mathcal {O}}_{L,{\mathfrak {P}}}/{\mathfrak {P}}}
35:
in a series of papers (1924; 1927; 1930), is a general theorem in
3846:
The formulation of the Artin reciprocity law as an equality of
4162:, Pure and Applied Mathematics, vol. 55, Academic Press,
3553:
An alternative version of the reciprocity law, leading to the
4313:, Graduate Texts in Mathematics, vol. 67, translated by
2792:{\displaystyle \sigma (\zeta _{m})=\zeta _{m}^{a_{\sigma }}.}
2331:≡ 1 (mod 4) or not. The Artin map is then defined on primes
1374:
1321:
3983:
3981:
3341:{\displaystyle (\mathbb {Z} /\ell \mathbb {Z} )^{\times }.}
1801:
2372:{\displaystyle p\mapsto \left({\frac {\Delta }{p}}\right)}
1088:
of I. Schur and used the reciprocity law to translate the
3286:{\displaystyle \operatorname {Gal} (F/\mathbb {Q} )=G/H.}
1505:{\displaystyle \left({\frac {L/K}{\mathfrak {p}}}\right)}
2992:{\displaystyle \ell ^{*}=(-1)^{\frac {\ell -1}{2}}\ell }
2548:> 1 be either an odd integer or a multiple of 4, let
3227:{\displaystyle G=\operatorname {Gal} (L/\mathbb {Q} ),}
2896:{\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }.}
2718:{\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }}
4081:
Emil Artin (1924) "Über eine neue Art von L-Reihen",
3850:-functions allows formulation of a generalisation to
3730:
3703:
3654:
3616:
3522:{\displaystyle \left({\frac {\ell ^{*}}{p}}\right)=1}
3487:
3361:
3303:
3240:
3189:
3125:
3077:
3008:
2940:
2920:
2858:
2808:
2741:
2683:
2641:
2589:
2554:
2507:
2461:
2427:
2388:
2344:
2303:
2257:
2216:
2185:
2155:
2099:
2031:
1846:
1589:
1554:
1471:
1431:
1370:
1317:
1288:
1256:
1220:
1196:
1164:
1131:
1026:
Artin's reciprocity law implies a description of the
994:
988:. Working out the cohomology groups establishes that
954:
808:
799:
in the sense of Artin and Tate. Then one proves that
719:
673:
653:
626:
606:
586:
566:
441:
405:
385:
365:
321:
295:
182:
152:
121:
85:
2670:{\displaystyle \operatorname {Gal} (L/\mathbb {Q} )}
2286:{\displaystyle \operatorname {Gal} (L/\mathbb {Q} )}
170:
is that there is a canonical isomorphism called the
3682:{\displaystyle \sigma :G\to \mathbb {C} ^{\times }}
3112:{\displaystyle F=\mathbb {Q} ({\sqrt {\ell ^{*}}})}
2293:can be identified with {±1}. The discriminant Δ of
1104:for an explanation of some of the terms used here)
3832:
3709:
3681:
3630:
3521:
3462:
3340:
3285:
3226:
3155:
3111:
3060:
2991:
2926:
2895:
2824:
2791:
2717:
2669:
2619:
2567:
2528:
2486:{\displaystyle \left({\frac {\Delta }{n}}\right).}
2485:
2443:
2409:
2371:
2311:
2285:
2243:
2202:
2167:
2129:
2051:
1979:
1807:
1572:
1504:
1457:
1409:
1356:
1303:
1266:
1230:
1206:
1174:
1139:
1000:
976:
937:
784:
686:
659:
639:
612:
592:
572:
546:
411:
391:
371:
347:
303:
278:
158:
134:
99:
2529:{\displaystyle \left({\frac {\Delta }{p}}\right)}
2410:{\displaystyle \left({\frac {\Delta }{p}}\right)}
67:. Artin's result provided a partial solution to
1837:such that the Artin map induces an isomorphism
1458:{\displaystyle \mathrm {Frob} _{\mathfrak {p}}}
4220:, Springer Monographs in Mathematics, Berlin:
3156:{\displaystyle L=\mathbb {Q} (\zeta _{\ell })}
2082:. The smallest defining modulus is called the
4373:(1967), "VII. Global class field theory", in
2934:be distinct odd primes. For convenience, let
694:are isomorphisms. This is the content of the
419:is defined by assembling the maps called the
8:
4385:, London: Academic Press, pp. 162–203,
4358:, London: Academic Press, pp. 128–161,
4346:(1967), "VI. Local class field theory", in
3987:
3293:Since the latter has order 2, the subgroup
2244:{\displaystyle L=\mathbb {Q} ({\sqrt {d}})}
4218:Reciprocity laws: From Euler to Eisenstein
4048:
4046:
2620:{\displaystyle L=\mathbb {Q} (\zeta _{m})}
3796:
3795:
3790:
3749:
3748:
3739:
3735:
3729:
3702:
3673:
3669:
3668:
3653:
3620:
3615:
3498:
3492:
3486:
3441:
3419:
3418:
3413:
3407:
3378:
3377:
3372:
3366:
3360:
3329:
3321:
3320:
3312:
3308:
3307:
3302:
3272:
3259:
3258:
3253:
3239:
3214:
3213:
3208:
3188:
3144:
3133:
3132:
3124:
3098:
3092:
3085:
3084:
3076:
3041:
3019:
3013:
3007:
2967:
2945:
2939:
2919:
2884:
2876:
2875:
2867:
2863:
2862:
2857:
2818:
2817:
2812:
2807:
2778:
2773:
2768:
2752:
2740:
2709:
2701:
2700:
2692:
2688:
2687:
2682:
2660:
2659:
2654:
2640:
2608:
2597:
2596:
2588:
2559:
2553:
2512:
2506:
2466:
2460:
2437:
2436:
2431:
2426:
2393:
2387:
2355:
2343:
2305:
2304:
2302:
2276:
2275:
2270:
2256:
2231:
2224:
2223:
2215:
2193:
2192:
2184:
2154:
2113:
2101:
2100:
2098:
2042:
2041:
2036:
2030:
1966:
1949:
1939:
1929:
1928:
1923:
1906:
1902:
1897:
1880:
1879:
1864:
1857:
1856:
1851:
1845:
1790:
1785:
1773:
1767:
1766:
1756:
1750:
1739:
1728:
1713:
1708:
1703:
1697:
1696:
1689:
1678:
1659:
1638:
1633:
1608:
1602:
1590:
1588:
1564:
1559:
1553:
1491:
1482:
1476:
1470:
1448:
1447:
1433:
1430:
1401:
1400:
1395:
1387:
1386:
1379:
1373:
1372:
1369:
1348:
1347:
1342:
1334:
1333:
1326:
1320:
1319:
1316:
1294:
1293:
1287:
1258:
1257:
1255:
1222:
1221:
1219:
1198:
1197:
1195:
1166:
1165:
1163:
1147:) has a concrete description in terms of
1133:
1132:
1130:
993:
968:
957:
956:
953:
925:
924:
910:
886:
875:
874:
861:
843:
822:
811:
810:
807:
773:
756:
742:
736:
718:
678:
672:
652:
631:
625:
605:
585:
565:
535:
519:
514:
499:
490:
484:
479:
470:
464:
459:
446:
440:
404:
384:
364:
348:{\displaystyle \operatorname {Gal} (L/K)}
334:
320:
296:
294:
267:
255:
230:
213:
209:
204:
199:
193:
181:
151:
126:
120:
89:
84:
4187:, vol. 110 (2 ed.), New York:
3862:
3572:(or Größencharakter) of a number field
1125:(such as a finite abelian extension of
667:-component of an idèle class. The maps
3561:associated to abelian extensions of a
2421:. More specifically, the conductor of
1107:The definition of the Artin map for a
1065:and to understand the behavior of the
4023:
4011:
3999:
2130:{\displaystyle {\mathfrak {f}}(L/K).}
1546:group of prime-to-Δ fractional ideals
7:
3638:be an abelian Galois extension with
2836:)∞, and the Artin map on a prime-to-
2052:{\displaystyle I_{L}^{\mathbf {c} }}
39:that forms a central part of global
3449:
2455:) is given by the Kronecker symbol
2102:
1768:
1698:
1492:
1449:
1402:
1388:
1349:
1335:
1295:
1259:
1223:
1199:
1167:
3809:
3806:
3803:
3800:
3797:
3762:
3759:
3756:
3753:
3750:
3697:), there exists a Hecke character
2514:
2468:
2395:
2357:
1956:
1953:
1950:
1898:
1639:
1565:
1443:
1440:
1437:
1434:
1096:Finite extensions of global fields
25:
4058:Automorphic forms on adèle groups
2906:Relation to quadratic reciprocity
1304:{\displaystyle D_{\mathfrak {p}}}
4317:, New York, Heidelberg, Berlin:
3588:interpreted Hecke characters as
3297:must be the group of squares in
2043:
1930:
1881:
1858:
3442:
2203:{\displaystyle K=\mathbb {Q} ,}
1573:{\displaystyle I_{K}^{\Delta }}
1282:, then the decomposition group
1267:{\displaystyle {\mathfrak {p}}}
1231:{\displaystyle {\mathfrak {p}}}
1207:{\displaystyle {\mathfrak {P}}}
1175:{\displaystyle {\mathfrak {p}}}
166:. One of the statements of the
3827:
3815:
3780:
3768:
3689:(i.e. one-dimensional complex
3664:
3453:
3443:
3431:
3425:
3390:
3384:
3326:
3304:
3263:
3247:
3218:
3202:
3150:
3137:
3106:
3089:
2964:
2954:
2881:
2859:
2825:{\displaystyle L/\mathbb {Q} }
2758:
2745:
2706:
2684:
2664:
2648:
2614:
2601:
2444:{\displaystyle L/\mathbb {Q} }
2348:
2280:
2264:
2238:
2228:
2121:
2107:
2017:is the norm map associated to
1974:
1960:
1941:
1936:
1916:
1893:
1872:
1721:
1667:
1653:
1644:
1061:in terms of the arithmetic of
977:{\displaystyle {\hat {H}}^{i}}
962:
929:
918:
904:
895:
880:
867:
851:
837:
828:
816:
779:
750:
729:
720:
528:
525:
507:
342:
328:
264:
249:
240:
236:
223:
1:
4185:Graduate Texts in Mathematics
3945:, Springer, 1992, Chapter VII
3874:History of Class Field Theory
1053:, it is used to describe the
3580:of the idèle class group of
2312:{\displaystyle \mathbb {Q} }
2059:is the fractional ideals of
1246:) since the latter group is
1140:{\displaystyle \mathbb {Q} }
1043:Hasse local–global principle
51:and the reciprocity laws of
31:, which was established by
687:{\displaystyle \theta _{v}}
640:{\displaystyle \theta _{v}}
620:is given by the local maps
304:{\displaystyle {\text{ab}}}
4423:
3943:Algebraische Zahlentheorie
3903:Algebraische Zahlentheorie
2568:{\displaystyle \zeta _{m}}
1079:Chebotarev density theorem
1015:
3594:reductive algebraic group
3119:and the cyclotomic field
1826:) states that there is a
49:quadratic reciprocity law
2493:This shows that a prime
2335:that do not divide Δ by
1534:(global) reciprocity map
1090:principalization problem
1077:, and also to prove the
1051:Takagi existence theorem
700:local class field theory
63:product formula for the
4276:Algebraic number theory
4181:Algebraic number theory
4160:Algebraic Number Fields
4158:Janusz, Gerald (1973),
3878:Algebraic Number Theory
3537:, i.e. if and only if,
2677:can be identified with
2168:{\displaystyle d\neq 1}
1001:{\displaystyle \theta }
613:{\displaystyle \theta }
412:{\displaystyle \theta }
69:Hilbert's ninth problem
3834:
3711:
3683:
3632:
3545:Statement in terms of
3541:is a square modulo ℓ.
3523:
3464:
3342:
3287:
3228:
3157:
3113:
3062:
2993:
2928:
2897:
2826:
2793:
2719:
2671:
2621:
2569:
2530:
2487:
2445:
2411:
2373:
2313:
2287:
2245:
2204:
2169:
2131:
2093:and typically denoted
2053:
1981:
1824:global reciprocity law
1809:
1744:
1694:
1574:
1506:
1459:
1411:
1358:
1305:
1268:
1232:
1208:
1176:
1141:
1041:which is based on the
1022:Eisenstein reciprocity
1002:
986:Tate cohomology groups
978:
939:
786:
688:
661:
641:
614:
594:
574:
548:
413:
393:
373:
349:
305:
280:
160:
136:
101:
4315:Greenberg, Marvin Jay
4254:Milne, James (2008),
3889:Neukirch (1999) p.391
3835:
3712:
3710:{\displaystyle \chi }
3684:
3633:
3524:
3465:
3343:
3288:
3229:
3158:
3114:
3063:
2994:
2929:
2927:{\displaystyle \ell }
2898:
2827:
2794:
2720:
2672:
2622:
2570:
2531:
2501:according to whether
2497:is split or inert in
2488:
2446:
2412:
2374:
2327:depending on whether
2314:
2288:
2246:
2205:
2170:
2132:
2073:defining modulus for
2054:
1982:
1820:Artin reciprocity law
1810:
1724:
1674:
1575:
1514:relative discriminant
1507:
1460:
1412:
1359:
1306:
1269:
1233:
1209:
1177:
1142:
1086:transfer homomorphism
1067:nonarchimedean places
1018:Quadratic reciprocity
1003:
979:
940:
787:
696:local reciprocity law
689:
662:
642:
615:
595:
575:
549:
425:local reciprocity map
414:
394:
374:
350:
306:
281:
168:Artin reciprocity law
161:
137:
135:{\displaystyle C_{L}}
102:
29:Artin reciprocity law
18:Artin reciprocity law
4128:, pp. 267–294,
3728:
3701:
3652:
3614:
3485:
3359:
3301:
3238:
3187:
3123:
3075:
3006:
2938:
2918:
2856:
2806:
2739:
2681:
2639:
2587:
2552:
2505:
2459:
2425:
2386:
2342:
2301:
2255:
2214:
2183:
2153:
2097:
2029:
1844:
1587:
1552:
1469:
1429:
1368:
1315:
1286:
1254:
1218:
1194:
1188:decomposition groups
1162:
1129:
1049:. Together with the
992:
952:
806:
717:
698:, a main theorem of
671:
651:
624:
604:
584:
564:
439:
403:
383:
363:
319:
293:
180:
150:
119:
83:
4054:Gelbart, Stephen S.
3814:
3767:
3631:{\displaystyle E/K}
3576:is defined to be a
3163:as follows. First,
2785:
2048:
1935:
1863:
1720:
1643:
1569:
1512:. If Δ denotes the
1045:and the use of the
1008:is an isomorphism.
524:
469:
430:norm residue symbol
100:{\displaystyle L/K}
4407:Class field theory
4344:Serre, Jean-Pierre
4305:Serre, Jean-Pierre
4257:Class field theory
4214:Lemmermeyer, Franz
4038:Class Field Theory
3972:10.1007/BF02941159
3932:Serre (1979) p.164
3923:Serre (1979) p.197
3914:Serre (1967) p.140
3830:
3786:
3731:
3707:
3679:
3628:
3519:
3481:, this shows that
3460:
3338:
3283:
3224:
3153:
3109:
3058:
2989:
2924:
2893:
2822:
2789:
2764:
2732:given by the rule
2715:
2667:
2617:
2565:
2536:is 1 or −1.
2526:
2483:
2441:
2407:
2369:
2309:
2283:
2241:
2200:
2177:squarefree integer
2165:
2127:
2049:
2032:
1977:
1919:
1847:
1805:
1800:
1695:
1629:
1570:
1555:
1544:is defined on the
1502:
1455:
1407:
1354:
1301:
1264:
1228:
1204:
1172:
1153:Frobenius elements
1137:
1055:abelian extensions
1047:Frobenius elements
998:
974:
935:
782:
764:
684:
657:
637:
610:
600:. More precisely,
590:
570:
544:
510:
455:
421:local Artin symbol
409:
389:
369:
345:
301:
276:
156:
132:
97:
41:class field theory
4231:978-3-540-66957-9
4198:978-0-387-94225-4
4135:978-3-540-43826-7
3941:Jürgen Neukirch,
3590:automorphic forms
3559:Artin L-functions
3555:Langlands program
3507:
3435:
3394:
3167:is a subfield of
3104:
3049:
3028:
2983:
2802:The conductor of
2540:Cyclotomic fields
2520:
2474:
2401:
2363:
2236:
2067:. Such a modulus
1947:
1779:
1620:
1496:
1238:are equal in Gal(
1112:abelian extension
1071:Artin L-functions
965:
883:
819:
757:
739:
660:{\displaystyle v}
593:{\displaystyle K}
573:{\displaystyle v}
538:
392:{\displaystyle K}
372:{\displaystyle L}
299:
270:
172:global symbol map
159:{\displaystyle L}
144:idèle class group
16:(Redirected from
4414:
4393:
4366:
4339:
4300:
4272:Neukirch, Jürgen
4267:
4266:
4265:
4250:
4209:
4172:
4154:
4113:Collected Papers
4102:Collected Papers
4087:Collected Papers
4070:
4068:
4050:
4041:
4033:
4027:
4021:
4015:
4009:
4003:
3997:
3991:
3988:Lemmermeyer 2000
3985:
3976:
3974:
3952:
3946:
3939:
3933:
3930:
3924:
3921:
3915:
3912:
3906:
3896:
3890:
3887:
3881:
3867:
3839:
3837:
3836:
3831:
3813:
3812:
3794:
3766:
3765:
3747:
3743:
3716:
3714:
3713:
3708:
3688:
3686:
3685:
3680:
3678:
3677:
3672:
3637:
3635:
3634:
3629:
3624:
3586:Robert Langlands
3529:if and only if,
3528:
3526:
3525:
3520:
3512:
3508:
3503:
3502:
3493:
3469:
3467:
3466:
3461:
3456:
3440:
3436:
3434:
3423:
3422:
3417:
3408:
3399:
3395:
3393:
3382:
3381:
3376:
3367:
3347:
3345:
3344:
3339:
3334:
3333:
3324:
3316:
3311:
3292:
3290:
3289:
3284:
3276:
3262:
3257:
3233:
3231:
3230:
3225:
3217:
3212:
3162:
3160:
3159:
3154:
3149:
3148:
3136:
3118:
3116:
3115:
3110:
3105:
3103:
3102:
3093:
3088:
3067:
3065:
3064:
3059:
3054:
3050:
3042:
3033:
3029:
3024:
3023:
3014:
2998:
2996:
2995:
2990:
2985:
2984:
2979:
2968:
2950:
2949:
2933:
2931:
2930:
2925:
2902:
2900:
2899:
2894:
2889:
2888:
2879:
2871:
2866:
2831:
2829:
2828:
2823:
2821:
2816:
2798:
2796:
2795:
2790:
2784:
2783:
2782:
2772:
2757:
2756:
2725:by sending σ to
2724:
2722:
2721:
2716:
2714:
2713:
2704:
2696:
2691:
2676:
2674:
2673:
2668:
2663:
2658:
2633:cyclotomic field
2626:
2624:
2623:
2618:
2613:
2612:
2600:
2581:th root of unity
2574:
2572:
2571:
2566:
2564:
2563:
2535:
2533:
2532:
2527:
2525:
2521:
2513:
2492:
2490:
2489:
2484:
2479:
2475:
2467:
2450:
2448:
2447:
2442:
2440:
2435:
2419:Kronecker symbol
2416:
2414:
2413:
2408:
2406:
2402:
2394:
2378:
2376:
2375:
2370:
2368:
2364:
2356:
2318:
2316:
2315:
2310:
2308:
2292:
2290:
2289:
2284:
2279:
2274:
2250:
2248:
2247:
2242:
2237:
2232:
2227:
2209:
2207:
2206:
2201:
2196:
2174:
2172:
2171:
2166:
2145:Quadratic fields
2136:
2134:
2133:
2128:
2117:
2106:
2105:
2058:
2056:
2055:
2050:
2047:
2046:
2040:
1986:
1984:
1983:
1978:
1970:
1959:
1948:
1940:
1934:
1933:
1927:
1915:
1914:
1910:
1901:
1892:
1891:
1884:
1868:
1862:
1861:
1855:
1814:
1812:
1811:
1806:
1804:
1803:
1797:
1796:
1795:
1794:
1784:
1780:
1778:
1777:
1772:
1771:
1764:
1760:
1751:
1743:
1738:
1719:
1718:
1717:
1707:
1702:
1701:
1693:
1688:
1663:
1642:
1637:
1625:
1621:
1616:
1612:
1603:
1580:, by linearity:
1579:
1577:
1576:
1571:
1568:
1563:
1511:
1509:
1508:
1503:
1501:
1497:
1495:
1490:
1486:
1477:
1464:
1462:
1461:
1456:
1454:
1453:
1452:
1446:
1416:
1414:
1413:
1408:
1406:
1405:
1399:
1394:
1393:
1392:
1391:
1378:
1377:
1363:
1361:
1360:
1355:
1353:
1352:
1346:
1341:
1340:
1339:
1338:
1325:
1324:
1310:
1308:
1307:
1302:
1300:
1299:
1298:
1273:
1271:
1270:
1265:
1263:
1262:
1237:
1235:
1234:
1229:
1227:
1226:
1213:
1211:
1210:
1205:
1203:
1202:
1181:
1179:
1178:
1173:
1171:
1170:
1146:
1144:
1143:
1138:
1136:
1030:of the absolute
1007:
1005:
1004:
999:
983:
981:
980:
975:
973:
972:
967:
966:
958:
944:
942:
941:
936:
928:
914:
894:
893:
885:
884:
876:
866:
865:
847:
827:
826:
821:
820:
812:
791:
789:
788:
783:
778:
777:
765:
746:
741:
740:
737:
693:
691:
690:
685:
683:
682:
666:
664:
663:
658:
646:
644:
643:
638:
636:
635:
619:
617:
616:
611:
599:
597:
596:
591:
579:
577:
576:
571:
553:
551:
550:
545:
540:
539:
536:
523:
518:
506:
505:
504:
503:
494:
489:
488:
474:
468:
463:
451:
450:
418:
416:
415:
410:
398:
396:
395:
390:
378:
376:
375:
370:
354:
352:
351:
346:
338:
315:of a group, and
310:
308:
307:
302:
300:
297:
285:
283:
282:
277:
272:
271:
268:
259:
239:
235:
234:
222:
221:
217:
203:
198:
197:
165:
163:
162:
157:
141:
139:
138:
133:
131:
130:
109:Galois extension
106:
104:
103:
98:
93:
21:
4422:
4421:
4417:
4416:
4415:
4413:
4412:
4411:
4397:
4396:
4375:Cassels, J.W.S.
4369:
4348:Cassels, J.W.S.
4342:
4329:
4319:Springer-Verlag
4303:
4290:
4280:Springer-Verlag
4270:
4263:
4261:
4260:(v4.0 ed.)
4253:
4232:
4222:Springer-Verlag
4212:
4199:
4189:Springer-Verlag
4175:
4170:
4157:
4136:
4126:Springer-Verlag
4118:
4093:(1965), 105–124
4078:
4073:
4052:
4051:
4044:
4034:
4030:
4022:
4018:
4010:
4006:
3998:
3994:
3986:
3979:
3954:
3953:
3949:
3940:
3936:
3931:
3927:
3922:
3918:
3913:
3909:
3899:Jürgen Neukirch
3897:
3893:
3888:
3884:
3868:
3864:
3860:
3726:
3725:
3699:
3698:
3667:
3650:
3649:
3645:. Then for any
3612:
3611:
3570:Hecke character
3551:
3533:modulo ℓ is in
3494:
3488:
3483:
3482:
3424:
3409:
3403:
3383:
3368:
3362:
3357:
3356:
3325:
3299:
3298:
3236:
3235:
3185:
3184:
3140:
3121:
3120:
3094:
3073:
3072:
3037:
3015:
3009:
3004:
3003:
2969:
2963:
2941:
2936:
2935:
2916:
2915:
2908:
2880:
2854:
2853:
2804:
2803:
2774:
2748:
2737:
2736:
2731:
2705:
2679:
2678:
2637:
2636:
2604:
2585:
2584:
2555:
2550:
2549:
2542:
2508:
2503:
2502:
2462:
2457:
2456:
2423:
2422:
2389:
2384:
2383:
2351:
2340:
2339:
2299:
2298:
2253:
2252:
2212:
2211:
2181:
2180:
2151:
2150:
2147:
2142:
2095:
2094:
2027:
2026:
2016:
1999:
1896:
1875:
1842:
1841:
1799:
1798:
1786:
1765:
1752:
1746:
1745:
1709:
1671:
1670:
1604:
1598:
1591:
1585:
1584:
1550:
1549:
1478:
1472:
1467:
1466:
1432:
1427:
1426:
1371:
1366:
1365:
1318:
1313:
1312:
1289:
1284:
1283:
1252:
1251:
1216:
1215:
1192:
1191:
1160:
1159:
1127:
1126:
1098:
1024:
1014:
990:
989:
955:
950:
949:
873:
857:
809:
804:
803:
797:class formation
769:
732:
715:
714:
708:
674:
669:
668:
649:
648:
627:
622:
621:
602:
601:
582:
581:
562:
561:
531:
495:
480:
475:
442:
437:
436:
401:
400:
381:
380:
361:
360:
317:
316:
291:
290:
263:
226:
205:
189:
178:
177:
148:
147:
122:
117:
116:
81:
80:
77:
45:reciprocity law
23:
22:
15:
12:
11:
5:
4420:
4418:
4410:
4409:
4399:
4398:
4395:
4394:
4367:
4340:
4327:
4301:
4288:
4268:
4251:
4230:
4210:
4197:
4173:
4168:
4155:
4134:
4116:
4105:
4094:
4091:Addison Wesley
4077:
4074:
4072:
4071:
4042:
4028:
4016:
4014:, example 3.10
4004:
4002:, example 3.11
3992:
3977:
3947:
3934:
3925:
3916:
3907:
3891:
3882:
3861:
3859:
3856:
3841:
3840:
3829:
3826:
3823:
3820:
3817:
3811:
3808:
3805:
3802:
3799:
3793:
3789:
3785:
3782:
3779:
3776:
3773:
3770:
3764:
3761:
3758:
3755:
3752:
3746:
3742:
3738:
3734:
3706:
3691:representation
3676:
3671:
3666:
3663:
3660:
3657:
3627:
3623:
3619:
3601:ring of adeles
3578:quasicharacter
3550:
3543:
3518:
3515:
3511:
3506:
3501:
3497:
3491:
3471:
3470:
3459:
3455:
3452:
3448:
3445:
3439:
3433:
3430:
3427:
3421:
3416:
3412:
3406:
3402:
3398:
3392:
3389:
3386:
3380:
3375:
3371:
3365:
3337:
3332:
3328:
3323:
3319:
3315:
3310:
3306:
3282:
3279:
3275:
3271:
3268:
3265:
3261:
3256:
3252:
3249:
3246:
3243:
3223:
3220:
3216:
3211:
3207:
3204:
3201:
3198:
3195:
3192:
3152:
3147:
3143:
3139:
3135:
3131:
3128:
3108:
3101:
3097:
3091:
3087:
3083:
3080:
3069:
3068:
3057:
3053:
3048:
3045:
3040:
3036:
3032:
3027:
3022:
3018:
3012:
2988:
2982:
2978:
2975:
2972:
2966:
2962:
2959:
2956:
2953:
2948:
2944:
2923:
2907:
2904:
2892:
2887:
2883:
2878:
2874:
2870:
2865:
2861:
2820:
2815:
2811:
2800:
2799:
2788:
2781:
2777:
2771:
2767:
2763:
2760:
2755:
2751:
2747:
2744:
2729:
2712:
2708:
2703:
2699:
2695:
2690:
2686:
2666:
2662:
2657:
2653:
2650:
2647:
2644:
2616:
2611:
2607:
2603:
2599:
2595:
2592:
2562:
2558:
2541:
2538:
2524:
2519:
2516:
2511:
2482:
2478:
2473:
2470:
2465:
2439:
2434:
2430:
2405:
2400:
2397:
2392:
2380:
2379:
2367:
2362:
2359:
2354:
2350:
2347:
2307:
2282:
2278:
2273:
2269:
2266:
2263:
2260:
2240:
2235:
2230:
2226:
2222:
2219:
2199:
2195:
2191:
2188:
2164:
2161:
2158:
2146:
2143:
2141:
2138:
2126:
2123:
2120:
2116:
2112:
2109:
2104:
2045:
2039:
2035:
2008:
1994:
1988:
1987:
1976:
1973:
1969:
1965:
1962:
1958:
1955:
1952:
1946:
1943:
1938:
1932:
1926:
1922:
1918:
1913:
1909:
1905:
1900:
1895:
1890:
1887:
1883:
1878:
1874:
1871:
1867:
1860:
1854:
1850:
1816:
1815:
1802:
1793:
1789:
1783:
1776:
1770:
1763:
1759:
1755:
1749:
1742:
1737:
1734:
1731:
1727:
1723:
1716:
1712:
1706:
1700:
1692:
1687:
1684:
1681:
1677:
1673:
1672:
1669:
1666:
1662:
1658:
1655:
1652:
1649:
1646:
1641:
1636:
1632:
1628:
1624:
1619:
1615:
1611:
1607:
1601:
1597:
1596:
1594:
1567:
1562:
1558:
1500:
1494:
1489:
1485:
1481:
1475:
1451:
1445:
1442:
1439:
1436:
1404:
1398:
1390:
1385:
1382:
1376:
1351:
1345:
1337:
1332:
1329:
1323:
1297:
1292:
1261:
1225:
1201:
1182:is a prime of
1169:
1135:
1097:
1094:
1028:abelianization
1013:
1010:
997:
971:
964:
961:
946:
945:
934:
931:
927:
923:
920:
917:
913:
909:
906:
903:
900:
897:
892:
889:
882:
879:
872:
869:
864:
860:
856:
853:
850:
846:
842:
839:
836:
833:
830:
825:
818:
815:
795:constitutes a
793:
792:
781:
776:
772:
768:
763:
760:
755:
752:
749:
745:
735:
731:
728:
725:
722:
707:
704:
681:
677:
656:
634:
630:
609:
589:
569:
557:for different
555:
554:
543:
534:
530:
527:
522:
517:
513:
509:
502:
498:
493:
487:
483:
478:
473:
467:
462:
458:
454:
449:
445:
408:
388:
368:
344:
341:
337:
333:
330:
327:
324:
313:abelianization
287:
286:
275:
266:
262:
258:
254:
251:
248:
245:
242:
238:
233:
229:
225:
220:
216:
212:
208:
202:
196:
192:
188:
185:
155:
142:stand for the
129:
125:
96:
92:
88:
76:
73:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4419:
4408:
4405:
4404:
4402:
4392:
4388:
4384:
4380:
4376:
4372:
4368:
4365:
4361:
4357:
4353:
4349:
4345:
4341:
4338:
4334:
4330:
4328:3-540-90424-7
4324:
4320:
4316:
4312:
4311:
4306:
4302:
4299:
4295:
4291:
4289:3-540-65399-6
4285:
4281:
4277:
4273:
4269:
4259:
4258:
4252:
4249:
4245:
4241:
4237:
4233:
4227:
4223:
4219:
4215:
4211:
4208:
4204:
4200:
4194:
4190:
4186:
4182:
4178:
4174:
4171:
4169:0-12-380250-4
4165:
4161:
4156:
4153:
4149:
4145:
4141:
4137:
4131:
4127:
4123:
4117:
4114:
4110:
4106:
4103:
4099:
4095:
4092:
4088:
4084:
4080:
4079:
4075:
4067:
4063:
4059:
4055:
4049:
4047:
4043:
4040:
4039:
4035:James Milne,
4032:
4029:
4026:, example 3.2
4025:
4020:
4017:
4013:
4008:
4005:
4001:
3996:
3993:
3989:
3984:
3982:
3978:
3973:
3969:
3965:
3961:
3957:
3951:
3948:
3944:
3938:
3935:
3929:
3926:
3920:
3917:
3911:
3908:
3904:
3900:
3895:
3892:
3886:
3883:
3879:
3875:
3871:
3866:
3863:
3857:
3855:
3853:
3849:
3844:
3824:
3821:
3818:
3791:
3787:
3783:
3777:
3774:
3771:
3744:
3740:
3736:
3732:
3724:
3723:
3722:
3720:
3704:
3696:
3693:of the group
3692:
3674:
3661:
3658:
3655:
3648:
3644:
3641:
3625:
3621:
3617:
3608:
3606:
3602:
3599:(1) over the
3598:
3595:
3591:
3587:
3583:
3579:
3575:
3571:
3566:
3564:
3560:
3556:
3548:
3544:
3542:
3540:
3536:
3532:
3516:
3513:
3509:
3504:
3499:
3495:
3489:
3480:
3476:
3457:
3450:
3446:
3437:
3428:
3414:
3410:
3404:
3400:
3396:
3387:
3373:
3369:
3363:
3355:
3354:
3353:
3351:
3335:
3330:
3317:
3313:
3296:
3280:
3277:
3273:
3269:
3266:
3254:
3250:
3244:
3241:
3221:
3209:
3205:
3199:
3196:
3193:
3190:
3182:
3178:
3174:
3170:
3166:
3145:
3141:
3129:
3126:
3099:
3095:
3081:
3078:
3055:
3051:
3046:
3043:
3038:
3034:
3030:
3025:
3020:
3016:
3010:
3002:
3001:
3000:
2986:
2980:
2976:
2973:
2970:
2960:
2957:
2951:
2946:
2942:
2921:
2913:
2905:
2903:
2890:
2885:
2872:
2868:
2851:
2847:
2843:
2839:
2835:
2813:
2809:
2786:
2779:
2775:
2769:
2765:
2761:
2753:
2749:
2742:
2735:
2734:
2733:
2728:
2710:
2697:
2693:
2655:
2651:
2645:
2642:
2634:
2630:
2609:
2605:
2593:
2590:
2582:
2580:
2560:
2556:
2547:
2539:
2537:
2522:
2517:
2509:
2500:
2496:
2480:
2476:
2471:
2463:
2454:
2432:
2428:
2420:
2403:
2398:
2390:
2365:
2360:
2352:
2345:
2338:
2337:
2336:
2334:
2330:
2326:
2322:
2296:
2271:
2267:
2261:
2258:
2233:
2220:
2217:
2197:
2189:
2186:
2178:
2162:
2159:
2156:
2144:
2139:
2137:
2124:
2118:
2114:
2110:
2092:
2091:
2087:
2084:conductor of
2081:
2080:
2076:
2070:
2066:
2062:
2037:
2033:
2024:
2020:
2015:
2011:
2006:
2005:
1997:
1993:
1971:
1967:
1963:
1944:
1924:
1920:
1911:
1907:
1903:
1888:
1885:
1876:
1869:
1865:
1852:
1848:
1840:
1839:
1838:
1836:
1832:
1829:
1825:
1821:
1791:
1787:
1781:
1774:
1761:
1757:
1753:
1747:
1740:
1735:
1732:
1729:
1725:
1714:
1710:
1704:
1690:
1685:
1682:
1679:
1675:
1664:
1660:
1656:
1650:
1647:
1634:
1630:
1626:
1622:
1617:
1613:
1609:
1605:
1599:
1592:
1583:
1582:
1581:
1560:
1556:
1547:
1543:
1539:
1535:
1531:
1527:
1523:
1519:
1515:
1498:
1487:
1483:
1479:
1473:
1425:) denoted by
1424:
1420:
1396:
1383:
1380:
1343:
1330:
1327:
1290:
1281:
1277:
1249:
1245:
1241:
1189:
1185:
1156:
1154:
1150:
1124:
1123:global fields
1120:
1116:
1113:
1110:
1105:
1103:
1095:
1093:
1091:
1087:
1082:
1080:
1076:
1072:
1068:
1064:
1060:
1056:
1052:
1048:
1044:
1040:
1037:
1033:
1029:
1023:
1019:
1011:
1009:
995:
987:
969:
959:
932:
921:
915:
911:
907:
901:
898:
890:
887:
877:
870:
862:
858:
854:
848:
844:
840:
834:
831:
823:
813:
802:
801:
800:
798:
774:
770:
766:
761:
758:
753:
747:
743:
733:
726:
723:
713:
712:
711:
705:
703:
701:
697:
679:
675:
654:
632:
628:
607:
587:
567:
560:
541:
532:
520:
515:
511:
500:
496:
491:
485:
481:
476:
471:
465:
460:
456:
452:
447:
443:
435:
434:
433:
432:
431:
426:
422:
406:
386:
366:
358:
339:
335:
331:
325:
322:
314:
273:
260:
256:
252:
246:
243:
231:
227:
218:
214:
210:
206:
200:
194:
190:
186:
183:
176:
175:
174:
173:
169:
153:
145:
127:
123:
114:
113:global fields
110:
94:
90:
86:
74:
72:
70:
66:
62:
58:
54:
50:
46:
42:
38:
37:number theory
34:
30:
19:
4382:
4379:Fröhlich, A.
4355:
4352:Fröhlich, A.
4310:Local Fields
4308:
4275:
4262:, retrieved
4256:
4217:
4180:
4159:
4121:
4112:
4108:
4101:
4100:5: 353–363;
4097:
4086:
4057:
4037:
4031:
4019:
4007:
3995:
3966:(1): 46–51,
3963:
3959:
3950:
3942:
3937:
3928:
3919:
3910:
3902:
3894:
3885:
3877:
3873:
3870:Helmut Hasse
3865:
3851:
3847:
3845:
3842:
3718:
3694:
3642:
3640:Galois group
3609:
3604:
3596:
3581:
3573:
3567:
3563:number field
3552:
3546:
3538:
3534:
3530:
3478:
3474:
3472:
3349:
3294:
3180:
3176:
3172:
3168:
3164:
3070:
2911:
2909:
2849:
2845:
2844:) is simply
2841:
2837:
2833:
2801:
2726:
2628:
2578:
2545:
2543:
2498:
2494:
2452:
2381:
2332:
2328:
2324:
2320:
2294:
2148:
2089:
2085:
2078:
2074:
2072:
2071:is called a
2068:
2064:
2060:
2022:
2018:
2013:
2009:
2003:
1995:
1991:
1989:
1834:
1830:
1823:
1819:
1817:
1541:
1537:
1533:
1529:
1526:Artin symbol
1525:
1521:
1517:
1422:
1418:
1279:
1243:
1239:
1183:
1157:
1149:prime ideals
1118:
1114:
1106:
1099:
1083:
1062:
1058:
1038:
1036:global field
1032:Galois group
1025:
1012:Significance
947:
794:
709:
695:
556:
428:
424:
420:
357:Galois group
311:denotes the
288:
171:
167:
78:
43:. The term "
28:
26:
4177:Lang, Serge
4085:3: 89–108;
3956:Artin, Emil
3557:, connects
2002:ray modulo
1075:meromorphic
984:denote the
399:. The map
65:norm symbol
4391:0153.07403
4371:Tate, John
4364:0153.07403
4337:0423.12016
4298:0956.11021
4264:2010-02-22
4248:0949.11002
4152:1065.11001
4124:, Berlin:
4111:7: 46–51;
4076:References
4024:Milne 2008
4012:Milne 2008
4000:Milne 2008
3721:such that
3549:-functions
2583:, and let
2577:primitive
1276:unramified
1190:of primes
1016:See also:
53:Eisenstein
33:Emil Artin
4115:, 159–164
4104:, 131–141
3819:χ
3772:σ
3705:χ
3675:×
3665:→
3656:σ
3647:character
3500:∗
3496:ℓ
3331:×
3318:ℓ
3245:
3200:
3146:ℓ
3142:ζ
3100:∗
3096:ℓ
3047:ℓ
3021:∗
3017:ℓ
2987:ℓ
2974:−
2971:ℓ
2958:−
2947:∗
2943:ℓ
2922:ℓ
2886:×
2780:σ
2766:ζ
2750:ζ
2743:σ
2711:×
2646:
2606:ζ
2557:ζ
2515:Δ
2469:Δ
2396:Δ
2358:Δ
2349:↦
2262:
2160:≠
2063:prime to
1945:∼
1942:⟶
1726:∏
1722:⟼
1676:∏
1651:
1645:⟶
1640:Δ
1618:⋅
1566:Δ
1530:Artin map
1186:then the
996:θ
963:^
902:
888:−
881:^
871:≃
835:
817:^
767:
762:→
727:
676:θ
629:θ
608:θ
529:→
521:×
466:×
444:θ
407:θ
326:
247:
241:→
184:θ
75:Statement
61:Hilbert's
4401:Category
4381:(eds.),
4354:(eds.),
4307:(1979),
4274:(1999),
4216:(2000),
4179:(1994),
4056:(1975),
3171:, so if
2140:Examples
4240:1761696
4207:1282723
4144:2077576
4066:0379375
3592:on the
2840:ideal (
2627:be the
2417:is the
2251:, then
2000:is the
1828:modulus
1248:abelian
647:on the
427:or the
355:is the
4389:
4362:
4335:
4325:
4296:
4286:
4246:
4238:
4228:
4205:
4195:
4166:
4150:
4142:
4132:
4064:
3990:, §3.2
3183:) and
3175:= Gal(
2382:where
1990:where
1524:, the
1214:above
1109:finite
948:where
559:places
423:, the
289:where
57:Kummer
3876:, in
3858:Notes
3473:When
3234:then
2852:) in
2848:(mod
2575:be a
2297:over
2175:is a
1536:) of
1532:, or
1364:over
1250:. If
1100:(See
1034:of a
706:Proof
379:over
107:be a
4323:ISBN
4284:ISBN
4226:ISBN
4193:ISBN
4164:ISBN
4130:ISBN
3610:Let
2914:and
2910:Let
2832:is (
2544:Let
2323:or 4
2210:and
2025:and
1822:(or
1818:The
1528:(or
1151:and
1073:are
1020:and
115:and
79:Let
55:and
27:The
4387:Zbl
4360:Zbl
4333:Zbl
4294:Zbl
4244:Zbl
4148:Zbl
3968:doi
3717:of
3603:of
3447:mod
3242:Gal
3197:Gal
2643:Gal
2631:th
2319:is
2259:Gal
2149:If
2007:, N
1833:of
1648:Gal
1516:of
1465:or
1278:in
1274:is
1158:If
1121:of
1057:of
899:Gal
832:Gal
759:lim
738:sep
724:Gal
580:of
359:of
323:Gal
244:Gal
146:of
111:of
59:to
4403::
4377:;
4350:;
4331:,
4321:,
4292:,
4282:,
4242:,
4236:MR
4234:,
4224:,
4203:MR
4201:,
4191:,
4183:,
4146:,
4140:MR
4138:,
4089:,
4062:MR
4045:^
3980:^
3962:,
3901:,
3872:,
3607:.
3597:GL
3584:.
3568:A
3477:=
3352:)
2635:.
2179:,
1998:,1
1548:,
1155:.
1081:.
702:.
537:ab
298:ab
269:ab
71:.
4069:.
3975:.
3970::
3964:7
3852:n
3848:L
3828:)
3825:s
3822:,
3816:(
3810:e
3807:k
3804:c
3801:e
3798:H
3792:K
3788:L
3784:=
3781:)
3778:s
3775:,
3769:(
3763:n
3760:i
3757:t
3754:r
3751:A
3745:K
3741:/
3737:E
3733:L
3719:K
3695:G
3670:C
3662:G
3659::
3643:G
3626:K
3622:/
3618:E
3605:K
3582:K
3574:K
3547:L
3539:p
3535:H
3531:p
3517:1
3514:=
3510:)
3505:p
3490:(
3479:p
3475:n
3458:.
3454:)
3451:H
3444:(
3438:)
3432:)
3429:n
3426:(
3420:Q
3415:/
3411:L
3405:(
3401:=
3397:)
3391:)
3388:n
3385:(
3379:Q
3374:/
3370:F
3364:(
3350:n
3336:.
3327:)
3322:Z
3314:/
3309:Z
3305:(
3295:H
3281:.
3278:H
3274:/
3270:G
3267:=
3264:)
3260:Q
3255:/
3251:F
3248:(
3222:,
3219:)
3215:Q
3210:/
3206:L
3203:(
3194:=
3191:G
3181:F
3179:/
3177:L
3173:H
3169:L
3165:F
3151:)
3138:(
3134:Q
3130:=
3127:L
3107:)
3090:(
3086:Q
3082:=
3079:F
3056:.
3052:)
3044:p
3039:(
3035:=
3031:)
3026:p
3011:(
2981:2
2977:1
2965:)
2961:1
2955:(
2952:=
2912:p
2891:.
2882:)
2877:Z
2873:m
2869:/
2864:Z
2860:(
2850:m
2846:n
2842:n
2838:m
2834:m
2819:Q
2814:/
2810:L
2787:.
2776:a
2770:m
2762:=
2759:)
2754:m
2746:(
2730:σ
2727:a
2707:)
2702:Z
2698:m
2694:/
2689:Z
2685:(
2665:)
2661:Q
2656:/
2652:L
2649:(
2629:m
2615:)
2610:m
2602:(
2598:Q
2594:=
2591:L
2579:m
2561:m
2546:m
2523:)
2518:p
2510:(
2499:L
2495:p
2481:.
2477:)
2472:n
2464:(
2453:n
2438:Q
2433:/
2429:L
2404:)
2399:p
2391:(
2366:)
2361:p
2353:(
2346:p
2333:p
2329:d
2325:d
2321:d
2306:Q
2295:L
2281:)
2277:Q
2272:/
2268:L
2265:(
2239:)
2234:d
2229:(
2225:Q
2221:=
2218:L
2198:,
2194:Q
2190:=
2187:K
2163:1
2157:d
2125:.
2122:)
2119:K
2115:/
2111:L
2108:(
2103:f
2090:K
2088:/
2086:L
2079:K
2077:/
2075:L
2069:c
2065:c
2061:L
2044:c
2038:L
2034:I
2023:K
2021:/
2019:L
2014:K
2012:/
2010:L
2004:c
1996:c
1992:K
1975:)
1972:K
1968:/
1964:L
1961:(
1957:l
1954:a
1951:G
1937:)
1931:c
1925:L
1921:I
1917:(
1912:K
1908:/
1904:L
1899:N
1894:)
1889:1
1886:,
1882:c
1877:K
1873:(
1870:i
1866:/
1859:c
1853:K
1849:I
1835:K
1831:c
1792:i
1788:n
1782:)
1775:i
1769:p
1762:K
1758:/
1754:L
1748:(
1741:m
1736:1
1733:=
1730:i
1715:i
1711:n
1705:i
1699:p
1691:m
1686:1
1683:=
1680:i
1668:)
1665:K
1661:/
1657:L
1654:(
1635:K
1631:I
1627::
1623:)
1614:K
1610:/
1606:L
1600:(
1593:{
1561:K
1557:I
1542:K
1540:/
1538:L
1522:K
1520:/
1518:L
1499:)
1493:p
1488:K
1484:/
1480:L
1474:(
1450:p
1444:b
1441:o
1438:r
1435:F
1423:K
1421:/
1419:L
1403:p
1397:/
1389:p
1384:,
1381:K
1375:O
1350:P
1344:/
1336:P
1331:,
1328:L
1322:O
1296:p
1291:D
1280:L
1260:p
1244:K
1242:/
1240:L
1224:p
1200:P
1184:K
1168:p
1134:Q
1119:K
1117:/
1115:L
1063:K
1059:K
1039:K
970:i
960:H
933:,
930:)
926:Z
922:,
919:)
916:K
912:/
908:L
905:(
896:(
891:2
878:H
868:)
863:L
859:C
855:,
852:)
849:K
845:/
841:L
838:(
829:(
824:0
814:H
780:)
775:L
771:C
754:,
751:)
748:K
744:/
734:K
730:(
721:(
680:v
655:v
633:v
588:K
568:v
542:,
533:G
526:)
516:v
512:L
508:(
501:v
497:K
492:/
486:v
482:L
477:N
472:/
461:v
457:K
453::
448:v
387:K
367:L
343:)
340:K
336:/
332:L
329:(
274:,
265:)
261:K
257:/
253:L
250:(
237:)
232:L
228:C
224:(
219:K
215:/
211:L
207:N
201:/
195:K
191:C
187::
154:L
128:L
124:C
95:K
91:/
87:L
20:)
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