230:
2315:
the two primes (2 + 3i) and (2 − 3i), so it cannot be in the decomposition group of either prime. The inertia group, being a subgroup of the decomposition group, is also the trivial group. There are two residue fields, one for each prime,
4367:
4232:
2197:
153:
3989:
3855:
3116:
1004:
1651:
1308:
3226:
2465:
3498:
349:
3355:
2797:
4096:
3736:
1355:
2042:
850:
670:
2383:
2620:
2696:
1964:
2306:
1897:
1558:
1058:
2649:
2719:
3633:
3609:
3573:
2548:
2520:
580:
4243:
4111:
2721:, which is certainly not divisible by 7. Therefore, the inertia group is the trivial group {1}. The Galois group of this residue field over the subfield
2058:
225:{\displaystyle {\begin{array}{ccc}O_{K}&\hookrightarrow &O_{L}\\\downarrow &&\downarrow \\K&\hookrightarrow &L\end{array}}}
3886:
3747:
3880:+ 1 is irreducible modulo 7. Therefore, there is only one prime factor, with inertia degree 2 and ramification index 1, and it is given by
3543:
available θ that satisfies the above hypotheses (see for example ). Therefore, the algorithm given above cannot be used to factor such
2243:
2211:
3017:
2388:
which are both isomorphic to the finite field with 13 elements. The
Frobenius element is the trivial automorphism; this means that
928:
2729:
has order 2, and is generated by the image of the
Frobenius element. The Frobenius element is none other than σ; this means that
1246:
3148:
2917:. The following procedure (Neukirch, p. 47) solves this problem in many cases. The strategy is to select an integer θ in
4496:
2394:
1581:
3423:
278:
3404:
The exceptional primes, for which the above result does not necessarily hold, are the ones not relatively prime to the
3282:
2735:
4016:
2939:
889:
4531:
4488:
4450:
3664:
1313:
1987:
922:; something that certainly need not be the case for extensions that are not Galois. The basic relations then read
4378:
809:
596:
3138:
is not one of finitely many exceptional primes (the precise condition is described below), the factorisation of
3636:
3741:
Therefore, there is only one prime factor, with inertia degree 1 and ramification index 2, and it is given by
1674:
to its
Frobenius and extending multiplicatively defines a homomorphism from the group of unramified ideals of
2322:
2568:
1706:
2654:
1912:
2935:
2252:
1717:
coincide. There, given a Galois ramified cover, all but finitely many points have the same number of
545:
4434:
4411:
1851:
1083:
1442:
which corresponds to the
Frobenius automorphism in the Galois group of the finite field extension
3405:
852:
70:
2311:
The decomposition groups in this case are both the trivial group {1}; indeed the automorphism σ
4536:
1969:
which is the finite field with two elements. The decomposition group must be equal to all of
1543:
800:
4506:
1799:
1016:
873:
4514:
8:
4415:
3504:
2628:
2554:, again because there is only one prime factor. However, this situation differs from the
262:
158:
2701:
3618:
3594:
3558:
2533:
2505:
1702:
1687:
1160:
are equal by basic Galois theory, it follows that the order of the decomposition group
4474:
4492:
3555:
Consider again the case of the
Gaussian integers. We take θ to be the imaginary unit
1791:
1407:
86:
4362:{\displaystyle Q_{2}=(13)\mathbf {Z} +(i-5)\mathbf {Z} =\cdots =(2-3i)\mathbf {Z} .}
4105:
prime factors, both with inertia degree and ramification index 1. They are given by
3385:
In the Galois case, the inertia degrees are all equal, and the ramification indices
2625:
which is the finite field with 7 = 49 elements. For example, the difference between
1724:
The splitting of primes in extensions that are not Galois may be studied by using a
4510:
4227:{\displaystyle Q_{1}=(13)\mathbf {Z} +(i+5)\mathbf {Z} =\cdots =(2+3i)\mathbf {Z} }
1698:
796:
140:
128:
49:
28:
4398:
4502:
1725:
269:
97:
need be considered, rather than two. This was certainly familiar before
Hilbert.
90:
1803:
1670:
we take. Furthermore, in the abelian case, associating an unramified prime of
4525:
3547:, and more sophisticated approaches must be used, such as that described in.
2192:{\displaystyle a+bi=2bi+a-bi=(1+i)\cdot (1-i)bi+a-bi\equiv a-bi{\bmod {1}}+i}
247:
78:
1740:
This section describes the splitting of prime ideals in the field extension
877:
804:
583:, hence the extension is unramified in all but finitely many prime ideals.
35:
21:
4465:
1729:
1728:
initially, i.e. a Galois extension that is somewhat larger. For example,
42:
17:
4455:
587:
3984:{\displaystyle Q=(7)\mathbf {Z} +(i^{2}+1)\mathbf {Z} =7\mathbf {Z} .}
3850:{\displaystyle Q=(2)\mathbf {Z} +(i+1)\mathbf {Z} =(1+i)\mathbf {Z} .}
1690:, which studies the finite abelian extensions of a given number field
2895:
Suppose that we wish to determine the factorisation of a prime ideal
4467:
A brief introduction to classical and adelic algebraic number theory
4451:"Splitting and ramification in number fields and Galois extensions"
1718:
1487:
is trivial, so the
Frobenius element is in this case an element of
1310:. It can be shown that this map is surjective, and it follows that
1657:
is an abelian group, the
Frobenius element of an unramified prime
3111:{\displaystyle h(X)=h_{1}(X)^{e_{1}}\cdots h_{n}(X)^{e_{n}},}
2780:
2439:
2174:
2019:
1794:. Although this case is far from representative — after all,
1732:
usually are 'regulated' by a degree 6 field containing them.
999:{\displaystyle pO_{L}=\left(\prod _{j=1}^{g}P_{j}\right)^{e}}
1804:
there aren't many quadratic fields with unique factorization
1303:{\displaystyle D_{P_{j}}\to \operatorname {Gal} (F_{j}/F)}
3515:
is from being the whole ring of integers (maximal order)
3221:{\displaystyle PO_{L}=Q_{1}^{e_{1}}\cdots Q_{n}^{e_{n}},}
2550:]. In this situation, the decomposition group is all of
93:, which is simpler in that only one kind of subgroup of
3639:
is the unit ideal, so there are no exceptional primes.
3258:
is equal to the degree of the corresponding polynomial
2460:{\displaystyle (a+bi)^{13}\equiv a+bi{\bmod {2}}\pm 3i}
1646:{\displaystyle \sigma D_{P_{j}}\sigma ^{-1}=D_{P_{j'}}}
272:
rings follows the existence of a unique decomposition
4246:
4114:
4019:
3889:
3750:
3667:
3621:
3597:
3561:
3527:
A significant caveat is that there exist examples of
3493:{\displaystyle \{y\in O_{L}:yO_{L}\subseteq O_{K}\};}
3426:
3285:
3151:
3020:
2738:
2704:
2657:
2631:
2571:
2536:
2508:
2397:
2325:
2255:
2061:
1990:
1915:
1854:
1584:
1546:
1316:
1249:
1019:
931:
812:
599:
281:
156:
1821:, and σ for the complex conjugation automorphism in
579:
is ramified in exactly those primes that divide the
344:{\displaystyle pO_{L}=\prod _{j=1}^{g}P_{j}^{e_{j}}}
4361:
4226:
4090:
3983:
3849:
3730:
3627:
3603:
3567:
3492:
3349:
3220:
3110:
2791:
2713:
2690:
2643:
2614:
2542:
2514:
2459:
2377:
2300:
2210:, since every prime that ramifies must divide the
2191:
2036:
1958:
1891:
1806:— it exhibits many of the features of the theory.
1735:
1645:
1552:
1349:
1302:
1052:
998:
844:
664:
343:
224:
3350:{\displaystyle Q_{j}=PO_{L}+h_{j}(\theta )O_{L},}
4523:
2953:; it is a monic polynomial with coefficients in
2792:{\displaystyle (a+bi)^{7}\equiv a-bi{\bmod {7}}}
113:be a finite extension of number fields, and let
4091:{\displaystyle X^{2}+1=(X+5)(X-5){\pmod {13}}.}
3654:, which amounts to factorising the polynomial
3130:are distinct monic irreducible polynomials in
2890:
75:splitting of prime ideals in Galois extensions
4484:Grundlehren der mathematischen Wissenschaften
3731:{\displaystyle X^{2}+1=(X+1)^{2}{\pmod {2}}.}
2934:by θ (such a θ is guaranteed to exist by the
1350:{\displaystyle \operatorname {Gal} (F_{j}/F)}
1185:This decomposition group contains a subgroup
4482:
4010:= (13). This time we have the factorisation
3484:
3427:
2037:{\displaystyle a+bi\equiv a-bi{\bmod {1}}+i}
139:, respectively, which are defined to be the
3417:. The conductor is defined to be the ideal
3267:, and there is an explicit formula for the
845:{\displaystyle G=\operatorname {Gal} (L/K)}
665:{\displaystyle =\sum _{j=1}^{g}e_{j}f_{j}.}
3249:. Furthermore, the inertia degree of each
1977:above 2. The inertia group is also all of
372:into a product of distinct maximal ideals
3872:≡ 3 mod 4. For concreteness we will take
3369:denotes here a lifting of the polynomial
2526:split. For example, (7) remains prime in
1902:The ramification index here is therefore
1220:that induce the identity automorphism on
60:factorise as products of prime ideals of
4473:
1410:goes further, to identify an element of
2378:{\displaystyle O_{L}/(2\pm 3i)O_{L}\ ,}
2244:Fermat's theorem on sums of two squares
782:
69:, provides one of the richest parts of
4524:
1455:. In the unramified case the order of
864:. That is, the prime ideal factors of
4463:
4430:
4417:Factoring Primes in Rings of Integers
4409:
2481:
2221:
1825:, there are three cases to consider.
85:. There is a geometric analogue, for
3646:= (2), we need to work in the field
3007:) factorises in the polynomial ring
4077:
3717:
3611:] is the whole ring of integers of
2562:act trivially on the residue field
1973:, since there is only one prime of
1385:and the order of the inertia group
544:. If this is the case then by the
13:
2238:into two distinct prime ideals in
1828:
496:. If it is bigger than 1 for some
14:
4548:
4443:
4412:"Essential Discriminant Divisors"
2862:Splits into two distinct factors
2615:{\displaystyle O_{L}/(7)O_{L}\ ,}
1212:, consisting of automorphisms of
1135:, is the subgroup of elements of
4343:
4302:
4270:
4211:
4170:
4138:
3965:
3945:
3906:
3831:
3799:
3767:
2691:{\displaystyle \sigma (1+i)=1-i}
1959:{\displaystyle O_{L}/(1+i)O_{L}}
787:In the following, the extension
4070:
3710:
2974:, we obtain a monic polynomial
2962:. Reducing the coefficients of
2301:{\displaystyle 13=(2+3i)(2-3i)}
1736:Example — the Gaussian integers
1697:In the geometric analogue, for
1523:are conjugate subgroups inside
1243:is the kernel of reduction map
1148:to itself. Since the degree of
1063:The relation above shows that /
4424:
4403:
4391:
4353:
4347:
4339:
4324:
4312:
4306:
4298:
4286:
4280:
4274:
4266:
4260:
4221:
4215:
4207:
4192:
4180:
4174:
4166:
4154:
4148:
4142:
4134:
4128:
4081:
4071:
4066:
4054:
4051:
4039:
4006:≡ 1 mod 4; we will again take
3975:
3969:
3955:
3949:
3941:
3922:
3916:
3910:
3902:
3896:
3841:
3835:
3827:
3815:
3809:
3803:
3795:
3783:
3777:
3771:
3763:
3757:
3721:
3711:
3700:
3687:
3481:
3475:
3331:
3325:
3089:
3082:
3053:
3046:
3030:
3024:
2755:
2739:
2673:
2661:
2593:
2587:
2414:
2398:
2356:
2341:
2295:
2280:
2277:
2262:
2134:
2122:
2116:
2104:
1943:
1931:
1880:
1867:
1861:
1855:
1501:, and thus also an element of
1344:
1323:
1297:
1276:
1267:
1086:this number is also equal to |
1032:
1020:
839:
825:
612:
600:
210:
198:
192:
173:
100:
1:
4384:
3550:
3240:are distinct prime ideals of
2986:, the (finite) residue field
2558:= 2 case, because now σ does
2242:; this is a manifestation of
1892:{\displaystyle (2)=(1+i)^{2}}
1686:, is a crucial ingredient of
268:From the basic theory of one-
239:be a non-zero prime ideal in
4379:Chebotarev's density theorem
890:unique factorisation theorem
524:, or that it is ramified in
20:, the interplay between the
7:
4372:
2938:), and then to examine the
2891:Computing the factorisation
77:is sometimes attributed to
10:
4553:
4479:Algebraische Zahlentheorie
4410:Stein, William A. (2002).
3575:, with minimal polynomial
2813:
1906:= 2. The residue field is
1707:algebraically closed field
147:in the field in question.
4487:. Vol. 322. Berlin:
2936:primitive element theorem
1682:. This map, known as the
1661:does not depend on which
1531:acts transitively on the
546:Chinese remainder theorem
4435:Method that Always Works
3503:it measures how far the
3142:has the following form:
1813:for the Galois group of
1084:orbit-stabilizer formula
803:can be used to show the
4532:Algebraic number theory
4464:Stein, William (2004),
2982:) with coefficients in
2206:prime that ramifies in
1553:{\displaystyle \sigma }
562:is a product of fields
448:residue field extension
71:algebraic number theory
4483:
4363:
4228:
4092:
3985:
3876:= (7). The polynomial
3851:
3732:
3629:
3605:
3569:
3494:
3351:
3222:
3112:
2844:Ramifies with index 2
2793:
2715:
2692:
2645:
2616:
2544:
2516:
2461:
2379:
2302:
2193:
2038:
1960:
1893:
1647:
1554:
1351:
1304:
1054:
1000:
974:
846:
666:
638:
500:, the field extension
408:naturally embeds into
381:, with multiplicities
345:
318:
250:, so that the residue
226:
4364:
4229:
4101:Therefore, there are
4093:
3986:
3860:The next case is for
3852:
3733:
3630:
3606:
3570:
3495:
3352:
3223:
3113:
2794:
2716:
2693:
2646:
2617:
2545:
2517:
2462:
2380:
2303:
2218:, which is −4.
2194:
2039:
1961:
1894:
1648:
1555:
1540:, one checks that if
1352:
1305:
1055:
1053:{\displaystyle =efg.}
1001:
954:
847:
801:prime avoidance lemma
667:
618:
581:relative discriminant
346:
298:
246:, or equivalently, a
227:
127:be the corresponding
4397:Milne, J.S. (2020).
4244:
4112:
4017:
3887:
3748:
3665:
3619:
3595:
3559:
3424:
3283:
3149:
3018:
2836:Decomposition group
2736:
2702:
2655:
2629:
2569:
2534:
2522:]; that is, it does
2506:
2395:
2323:
2253:
2059:
1988:
1913:
1852:
1800:unique factorisation
1582:
1544:
1314:
1247:
1071:of prime factors of
1017:
929:
888:. From this and the
810:
783:The Galois situation
597:
586:Multiplicativity of
279:
154:
4399:Class Field Theory.
3539:such that there is
3214:
3189:
3134:. Then, as long as
2644:{\displaystyle 1+i}
1748:. That is, we take
1711:decomposition group
1124:decomposition group
918:are independent of
795:is assumed to be a
739:ramifies completely
340:
4359:
4224:
4088:
3981:
3847:
3728:
3625:
3601:
3565:
3490:
3347:
3218:
3193:
3168:
3108:
2940:minimal polynomial
2930:is generated over
2789:
2714:{\displaystyle 2i}
2711:
2688:
2641:
2612:
2540:
2512:
2494:≡ 3 mod 4 remains
2457:
2375:
2298:
2202:In fact, 2 is the
2189:
2034:
1956:
1889:
1709:, the concepts of
1703:algebraic geometry
1688:class field theory
1643:
1550:
1406:The theory of the
1347:
1300:
1229:. In other words,
1067:equals the number
1050:
996:
892:, it follows that
842:
662:
481:ramification index
341:
319:
222:
220:
87:ramified coverings
41:, and the way the
4498:978-3-540-65399-8
3994:The last case is
3628:{\displaystyle i}
3604:{\displaystyle i}
3568:{\displaystyle i}
2888:
2887:
2827:How it splits in
2802:for any integers
2608:
2543:{\displaystyle i}
2515:{\displaystyle i}
2470:for any integers
2371:
2047:for any integers
1792:Gaussian integers
1699:complex manifolds
1527:: Recalling that
1408:Frobenius element
1357:is isomorphic to
1156:and the order of
853:acts transitively
767:= ), we say that
734:= ), we say that
706:splits completely
701:= ), we say that
470:The multiplicity
4544:
4518:
4486:
4475:Neukirch, Jürgen
4470:
4460:
4437:
4428:
4422:
4421:
4407:
4401:
4395:
4368:
4366:
4365:
4360:
4346:
4305:
4273:
4256:
4255:
4233:
4231:
4230:
4225:
4214:
4173:
4141:
4124:
4123:
4097:
4095:
4094:
4089:
4084:
4029:
4028:
3990:
3988:
3987:
3982:
3968:
3948:
3934:
3933:
3909:
3856:
3854:
3853:
3848:
3834:
3802:
3770:
3737:
3735:
3734:
3729:
3724:
3708:
3707:
3677:
3676:
3634:
3632:
3631:
3626:
3610:
3608:
3607:
3602:
3574:
3572:
3571:
3566:
3499:
3497:
3496:
3491:
3474:
3473:
3461:
3460:
3445:
3444:
3356:
3354:
3353:
3348:
3343:
3342:
3324:
3323:
3311:
3310:
3295:
3294:
3227:
3225:
3224:
3219:
3213:
3212:
3211:
3201:
3188:
3187:
3186:
3176:
3164:
3163:
3117:
3115:
3114:
3109:
3104:
3103:
3102:
3101:
3081:
3080:
3068:
3067:
3066:
3065:
3045:
3044:
2818:
2817:
2798:
2796:
2795:
2790:
2788:
2787:
2763:
2762:
2720:
2718:
2717:
2712:
2697:
2695:
2694:
2689:
2650:
2648:
2647:
2642:
2621:
2619:
2618:
2613:
2606:
2605:
2604:
2586:
2581:
2580:
2549:
2547:
2546:
2541:
2521:
2519:
2518:
2513:
2466:
2464:
2463:
2458:
2447:
2446:
2422:
2421:
2384:
2382:
2381:
2376:
2369:
2368:
2367:
2340:
2335:
2334:
2307:
2305:
2304:
2299:
2198:
2196:
2195:
2190:
2182:
2181:
2043:
2041:
2040:
2035:
2027:
2026:
1965:
1963:
1962:
1957:
1955:
1954:
1930:
1925:
1924:
1898:
1896:
1895:
1890:
1888:
1887:
1653:. Therefore, if
1652:
1650:
1649:
1644:
1642:
1641:
1640:
1639:
1638:
1617:
1616:
1604:
1603:
1602:
1601:
1559:
1557:
1556:
1551:
1356:
1354:
1353:
1348:
1340:
1335:
1334:
1309:
1307:
1306:
1301:
1293:
1288:
1287:
1266:
1265:
1264:
1263:
1139:sending a given
1059:
1057:
1056:
1051:
1005:
1003:
1002:
997:
995:
994:
989:
985:
984:
983:
973:
968:
944:
943:
851:
849:
848:
843:
835:
797:Galois extension
671:
669:
668:
663:
658:
657:
648:
647:
637:
632:
571:. The extension
516:(or we say that
350:
348:
347:
342:
339:
338:
337:
327:
317:
312:
294:
293:
231:
229:
228:
223:
221:
196:
187:
186:
170:
169:
143:of the integers
141:integral closure
129:ring of integers
91:Riemann surfaces
50:ring of integers
29:Galois extension
4552:
4551:
4547:
4546:
4545:
4543:
4542:
4541:
4522:
4521:
4499:
4489:Springer-Verlag
4449:
4446:
4441:
4440:
4429:
4425:
4408:
4404:
4396:
4392:
4387:
4375:
4342:
4301:
4269:
4251:
4247:
4245:
4242:
4241:
4210:
4169:
4137:
4119:
4115:
4113:
4110:
4109:
4069:
4024:
4020:
4018:
4015:
4014:
3964:
3944:
3929:
3925:
3905:
3888:
3885:
3884:
3830:
3798:
3766:
3749:
3746:
3745:
3709:
3703:
3699:
3672:
3668:
3666:
3663:
3662:
3620:
3617:
3616:
3596:
3593:
3592:
3560:
3557:
3556:
3553:
3523:
3514:
3469:
3465:
3456:
3452:
3440:
3436:
3425:
3422:
3421:
3416:
3401:are all equal.
3400:
3391:
3377:
3368:
3338:
3334:
3319:
3315:
3306:
3302:
3290:
3286:
3284:
3281:
3280:
3275:
3266:
3257:
3248:
3239:
3207:
3203:
3202:
3197:
3182:
3178:
3177:
3172:
3159:
3155:
3150:
3147:
3146:
3129:
3097:
3093:
3092:
3088:
3076:
3072:
3061:
3057:
3056:
3052:
3040:
3036:
3019:
3016:
3015:
2999:. Suppose that
2994:
2961:
2925:
2916:
2908:into primes of
2907:
2893:
2816:
2783:
2779:
2758:
2754:
2737:
2734:
2733:
2703:
2700:
2699:
2656:
2653:
2652:
2630:
2627:
2626:
2600:
2596:
2582:
2576:
2572:
2570:
2567:
2566:
2535:
2532:
2531:
2507:
2504:
2503:
2488:
2486:≡ 3 mod 4
2442:
2438:
2417:
2413:
2396:
2393:
2392:
2363:
2359:
2336:
2330:
2326:
2324:
2321:
2320:
2254:
2251:
2250:
2246:. For example:
2228:
2226:≡ 1 mod 4
2177:
2173:
2060:
2057:
2056:
2022:
2018:
1989:
1986:
1985:
1950:
1946:
1926:
1920:
1916:
1914:
1911:
1910:
1883:
1879:
1853:
1850:
1849:
1837:The prime 2 of
1835:
1790:is the ring of
1785:
1772:
1738:
1726:splitting field
1669:
1631:
1630:
1626:
1625:
1621:
1609:
1605:
1597:
1593:
1592:
1588:
1583:
1580:
1579:
1577:
1568:
1545:
1542:
1541:
1539:
1522:
1521:
1500:
1499:
1486:
1485:
1468:
1467:
1450:
1437:
1436:
1423:
1422:
1398:
1397:
1384:
1383:
1370:
1369:
1336:
1330:
1326:
1315:
1312:
1311:
1289:
1283:
1279:
1259:
1255:
1254:
1250:
1248:
1245:
1244:
1242:
1241:
1228:
1211:
1198:
1197:
1173:
1172:
1147:
1134:
1121:
1120:
1103:
1102:
1080:
1018:
1015:
1014:
990:
979:
975:
969:
958:
953:
949:
948:
939:
935:
930:
927:
926:
917:
904:
863:
831:
811:
808:
807:
785:
766:
757:
733:
724:
692:
683:
653:
649:
643:
639:
633:
622:
598:
595:
594:
570:
560:
553:
491:
478:
462:
445:
432:
422:
416:
402:
389:
380:
366:
359:
333:
329:
328:
323:
313:
302:
289:
285:
280:
277:
276:
255:
244:
219:
218:
213:
208:
202:
201:
195:
189:
188:
182:
178:
176:
171:
165:
161:
157:
155:
152:
151:
125:
118:
103:
68:
59:
12:
11:
5:
4550:
4540:
4539:
4534:
4520:
4519:
4497:
4471:
4461:
4445:
4444:External links
4442:
4439:
4438:
4423:
4402:
4389:
4388:
4386:
4383:
4382:
4381:
4374:
4371:
4370:
4369:
4358:
4355:
4352:
4349:
4345:
4341:
4338:
4335:
4332:
4329:
4326:
4323:
4320:
4317:
4314:
4311:
4308:
4304:
4300:
4297:
4294:
4291:
4288:
4285:
4282:
4279:
4276:
4272:
4268:
4265:
4262:
4259:
4254:
4250:
4235:
4234:
4223:
4220:
4217:
4213:
4209:
4206:
4203:
4200:
4197:
4194:
4191:
4188:
4185:
4182:
4179:
4176:
4172:
4168:
4165:
4162:
4159:
4156:
4153:
4150:
4147:
4144:
4140:
4136:
4133:
4130:
4127:
4122:
4118:
4099:
4098:
4087:
4083:
4080:
4076:
4073:
4068:
4065:
4062:
4059:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4035:
4032:
4027:
4023:
4002:) for a prime
3992:
3991:
3980:
3977:
3974:
3971:
3967:
3963:
3960:
3957:
3954:
3951:
3947:
3943:
3940:
3937:
3932:
3928:
3924:
3921:
3918:
3915:
3912:
3908:
3904:
3901:
3898:
3895:
3892:
3868:) for a prime
3858:
3857:
3846:
3843:
3840:
3837:
3833:
3829:
3826:
3823:
3820:
3817:
3814:
3811:
3808:
3805:
3801:
3797:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3769:
3765:
3762:
3759:
3756:
3753:
3739:
3738:
3727:
3723:
3720:
3716:
3713:
3706:
3702:
3698:
3695:
3692:
3689:
3686:
3683:
3680:
3675:
3671:
3658:+ 1 modulo 2:
3624:
3600:
3564:
3552:
3549:
3519:
3510:
3501:
3500:
3489:
3486:
3483:
3480:
3477:
3472:
3468:
3464:
3459:
3455:
3451:
3448:
3443:
3439:
3435:
3432:
3429:
3412:
3396:
3389:
3373:
3364:
3358:
3357:
3346:
3341:
3337:
3333:
3330:
3327:
3322:
3318:
3314:
3309:
3305:
3301:
3298:
3293:
3289:
3271:
3262:
3253:
3244:
3235:
3229:
3228:
3217:
3210:
3206:
3200:
3196:
3192:
3185:
3181:
3175:
3171:
3167:
3162:
3158:
3154:
3125:
3119:
3118:
3107:
3100:
3096:
3091:
3087:
3084:
3079:
3075:
3071:
3064:
3060:
3055:
3051:
3048:
3043:
3039:
3035:
3032:
3029:
3026:
3023:
2990:
2957:
2921:
2912:
2903:
2892:
2889:
2886:
2885:
2880:
2877:
2876:Remains inert
2874:
2870:
2869:
2866:
2863:
2860:
2856:
2855:
2850:
2845:
2842:
2838:
2837:
2834:
2833:Inertia group
2831:
2825:
2815:
2812:
2800:
2799:
2786:
2782:
2778:
2775:
2772:
2769:
2766:
2761:
2757:
2753:
2750:
2747:
2744:
2741:
2710:
2707:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2640:
2637:
2634:
2623:
2622:
2611:
2603:
2599:
2595:
2592:
2589:
2585:
2579:
2575:
2539:
2511:
2487:
2480:
2468:
2467:
2456:
2453:
2450:
2445:
2441:
2437:
2434:
2431:
2428:
2425:
2420:
2416:
2412:
2409:
2406:
2403:
2400:
2386:
2385:
2374:
2366:
2362:
2358:
2355:
2352:
2349:
2346:
2343:
2339:
2333:
2329:
2309:
2308:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2227:
2220:
2188:
2185:
2180:
2176:
2172:
2169:
2166:
2163:
2160:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2094:
2091:
2088:
2085:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2045:
2044:
2033:
2030:
2025:
2021:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1967:
1966:
1953:
1949:
1945:
1942:
1939:
1936:
1933:
1929:
1923:
1919:
1900:
1899:
1886:
1882:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1834:
1827:
1781:
1768:
1737:
1734:
1665:
1637:
1634:
1629:
1624:
1620:
1615:
1612:
1608:
1600:
1596:
1591:
1587:
1573:
1564:
1549:
1535:
1517:
1513:
1505:. For varying
1495:
1491:
1481:
1477:
1463:
1459:
1446:
1432:
1428:
1418:
1414:
1393:
1389:
1379:
1375:
1365:
1361:
1346:
1343:
1339:
1333:
1329:
1325:
1322:
1319:
1299:
1296:
1292:
1286:
1282:
1278:
1275:
1272:
1269:
1262:
1258:
1253:
1237:
1233:
1224:
1207:
1193:
1189:
1168:
1164:
1143:
1130:
1116:
1112:
1098:
1094:
1078:
1061:
1060:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1008:
1007:
993:
988:
982:
978:
972:
967:
964:
961:
957:
952:
947:
942:
938:
934:
913:
900:
872:form a single
859:
841:
838:
834:
830:
827:
824:
821:
818:
815:
784:
781:
762:
753:
745:. Finally, if
729:
720:
693:= 1 for every
688:
679:
673:
672:
661:
656:
652:
646:
642:
636:
631:
628:
625:
621:
617:
614:
611:
608:
605:
602:
566:
558:
551:
528:). Otherwise,
487:
474:
458:
452:inertia degree
441:
428:
420:
412:
400:
385:
376:
364:
357:
352:
351:
336:
332:
326:
322:
316:
311:
308:
305:
301:
297:
292:
288:
284:
253:
242:
233:
232:
217:
214:
212:
209:
207:
204:
203:
200:
197:
194:
191:
190:
185:
181:
177:
175:
172:
168:
164:
160:
159:
123:
116:
102:
99:
83:Hilbert theory
81:by calling it
64:
55:
9:
6:
4:
3:
2:
4549:
4538:
4537:Galois theory
4535:
4533:
4530:
4529:
4527:
4516:
4512:
4508:
4504:
4500:
4494:
4490:
4485:
4480:
4476:
4472:
4469:
4468:
4462:
4458:
4457:
4452:
4448:
4447:
4436:
4432:
4427:
4419:
4418:
4413:
4406:
4400:
4394:
4390:
4380:
4377:
4376:
4356:
4350:
4336:
4333:
4330:
4327:
4321:
4318:
4315:
4309:
4295:
4292:
4289:
4283:
4277:
4263:
4257:
4252:
4248:
4240:
4239:
4238:
4218:
4204:
4201:
4198:
4195:
4189:
4186:
4183:
4177:
4163:
4160:
4157:
4151:
4145:
4131:
4125:
4120:
4116:
4108:
4107:
4106:
4104:
4085:
4078:
4074:
4063:
4060:
4057:
4048:
4045:
4042:
4036:
4033:
4030:
4025:
4021:
4013:
4012:
4011:
4009:
4005:
4001:
3997:
3978:
3972:
3961:
3958:
3952:
3938:
3935:
3930:
3926:
3919:
3913:
3899:
3893:
3890:
3883:
3882:
3881:
3879:
3875:
3871:
3867:
3863:
3844:
3838:
3824:
3821:
3818:
3812:
3806:
3792:
3789:
3786:
3780:
3774:
3760:
3754:
3751:
3744:
3743:
3742:
3725:
3718:
3714:
3704:
3696:
3693:
3690:
3684:
3681:
3678:
3673:
3669:
3661:
3660:
3659:
3657:
3653:
3649:
3645:
3640:
3638:
3622:
3614:
3598:
3590:
3586:
3582:
3578:
3562:
3548:
3546:
3542:
3538:
3534:
3530:
3525:
3522:
3518:
3513:
3509:
3506:
3487:
3478:
3470:
3466:
3462:
3457:
3453:
3449:
3446:
3441:
3437:
3433:
3430:
3420:
3419:
3418:
3415:
3411:
3407:
3402:
3399:
3395:
3388:
3383:
3381:
3376:
3372:
3367:
3363:
3344:
3339:
3335:
3328:
3320:
3316:
3312:
3307:
3303:
3299:
3296:
3291:
3287:
3279:
3278:
3277:
3274:
3270:
3265:
3261:
3256:
3252:
3247:
3243:
3238:
3234:
3215:
3208:
3204:
3198:
3194:
3190:
3183:
3179:
3173:
3169:
3165:
3160:
3156:
3152:
3145:
3144:
3143:
3141:
3137:
3133:
3128:
3124:
3105:
3098:
3094:
3085:
3077:
3073:
3069:
3062:
3058:
3049:
3041:
3037:
3033:
3027:
3021:
3014:
3013:
3012:
3010:
3006:
3002:
2998:
2993:
2989:
2985:
2981:
2977:
2973:
2969:
2965:
2960:
2956:
2952:
2948:
2944:
2941:
2937:
2933:
2929:
2924:
2920:
2915:
2911:
2906:
2902:
2898:
2884:
2881:
2878:
2875:
2872:
2871:
2867:
2864:
2861:
2858:
2857:
2854:
2851:
2849:
2846:
2843:
2840:
2839:
2835:
2832:
2830:
2826:
2824:
2820:
2819:
2811:
2809:
2805:
2784:
2776:
2773:
2770:
2767:
2764:
2759:
2751:
2748:
2745:
2742:
2732:
2731:
2730:
2728:
2724:
2708:
2705:
2685:
2682:
2679:
2676:
2670:
2667:
2664:
2658:
2638:
2635:
2632:
2609:
2601:
2597:
2590:
2583:
2577:
2573:
2565:
2564:
2563:
2561:
2557:
2553:
2537:
2529:
2525:
2509:
2501:
2497:
2493:
2485:
2479:
2477:
2473:
2454:
2451:
2448:
2443:
2435:
2432:
2429:
2426:
2423:
2418:
2410:
2407:
2404:
2401:
2391:
2390:
2389:
2372:
2364:
2360:
2353:
2350:
2347:
2344:
2337:
2331:
2327:
2319:
2318:
2317:
2314:
2292:
2289:
2286:
2283:
2274:
2271:
2268:
2265:
2259:
2256:
2249:
2248:
2247:
2245:
2241:
2237:
2233:
2225:
2219:
2217:
2213:
2209:
2205:
2200:
2186:
2183:
2178:
2170:
2167:
2164:
2161:
2158:
2155:
2152:
2149:
2146:
2143:
2140:
2137:
2131:
2128:
2125:
2119:
2113:
2110:
2107:
2101:
2098:
2095:
2092:
2089:
2086:
2083:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2054:
2050:
2031:
2028:
2023:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1984:
1983:
1982:
1980:
1976:
1972:
1951:
1947:
1940:
1937:
1934:
1927:
1921:
1917:
1909:
1908:
1907:
1905:
1884:
1876:
1873:
1870:
1864:
1858:
1848:
1847:
1846:
1844:
1840:
1832:
1826:
1824:
1820:
1816:
1812:
1807:
1805:
1801:
1797:
1793:
1789:
1784:
1780:
1776:
1771:
1767:
1763:
1759:
1755:
1751:
1747:
1743:
1733:
1731:
1727:
1722:
1720:
1716:
1715:inertia group
1712:
1708:
1704:
1700:
1695:
1693:
1689:
1685:
1681:
1677:
1673:
1668:
1664:
1660:
1656:
1635:
1632:
1627:
1622:
1618:
1613:
1610:
1606:
1598:
1594:
1589:
1585:
1576:
1572:
1567:
1563:
1547:
1538:
1534:
1530:
1526:
1520:
1516:
1512:
1509:, the groups
1508:
1504:
1498:
1494:
1490:
1484:
1480:
1476:
1472:
1466:
1462:
1458:
1454:
1449:
1445:
1441:
1435:
1431:
1427:
1421:
1417:
1413:
1409:
1404:
1402:
1396:
1392:
1388:
1382:
1378:
1374:
1368:
1364:
1360:
1341:
1337:
1331:
1327:
1320:
1317:
1294:
1290:
1284:
1280:
1273:
1270:
1260:
1256:
1251:
1240:
1236:
1232:
1227:
1223:
1219:
1215:
1210:
1206:
1202:
1201:inertia group
1196:
1192:
1188:
1183:
1181:
1177:
1171:
1167:
1163:
1159:
1155:
1151:
1146:
1142:
1138:
1133:
1129:
1125:
1119:
1115:
1111:
1107:
1101:
1097:
1093:
1089:
1085:
1081:
1074:
1070:
1066:
1047:
1044:
1041:
1038:
1035:
1029:
1026:
1023:
1013:
1012:
1011:
991:
986:
980:
976:
970:
965:
962:
959:
955:
950:
945:
940:
936:
932:
925:
924:
923:
921:
916:
912:
908:
903:
899:
895:
891:
887:
883:
879:
878:automorphisms
875:
871:
867:
862:
858:
854:
836:
832:
828:
822:
819:
816:
813:
806:
802:
798:
794:
790:
780:
778:
774:
770:
765:
761:
756:
752:
748:
744:
740:
737:
732:
728:
723:
719:
715:
711:
707:
704:
700:
696:
691:
687:
682:
678:
659:
654:
650:
644:
640:
634:
629:
626:
623:
619:
615:
609:
606:
603:
593:
592:
591:
589:
584:
582:
578:
574:
569:
565:
561:
554:
548:the quotient
547:
543:
539:
535:
531:
527:
523:
519:
515:
511:
507:
503:
499:
495:
490:
486:
482:
477:
473:
468:
466:
461:
457:
453:
449:
444:
440:
437:, the degree
436:
431:
427:
423:
415:
411:
407:
403:
396:
391:
388:
384:
379:
375:
371:
367:
361:generated in
360:
354:of the ideal
334:
330:
324:
320:
314:
309:
306:
303:
299:
295:
290:
286:
282:
275:
274:
273:
271:
266:
264:
260:
256:
249:
248:maximal ideal
245:
238:
235:Finally, let
215:
205:
183:
179:
166:
162:
150:
149:
148:
146:
142:
138:
134:
130:
126:
119:
112:
108:
98:
96:
92:
88:
84:
80:
79:David Hilbert
76:
72:
67:
63:
58:
54:
51:
47:
44:
40:
37:
33:
30:
26:
23:
19:
4478:
4466:
4454:
4426:
4416:
4405:
4393:
4236:
4102:
4100:
4007:
4003:
3999:
3995:
3993:
3877:
3873:
3869:
3865:
3861:
3859:
3740:
3655:
3651:
3647:
3643:
3641:
3612:
3588:
3584:
3580:
3576:
3554:
3544:
3540:
3536:
3532:
3528:
3526:
3520:
3516:
3511:
3507:
3502:
3413:
3409:
3408:of the ring
3403:
3397:
3393:
3386:
3384:
3379:
3374:
3370:
3365:
3361:
3359:
3272:
3268:
3263:
3259:
3254:
3250:
3245:
3241:
3236:
3232:
3230:
3139:
3135:
3131:
3126:
3122:
3120:
3008:
3004:
3000:
2996:
2991:
2987:
2983:
2979:
2975:
2971:
2967:
2963:
2958:
2954:
2950:
2949:) of θ over
2946:
2942:
2931:
2927:
2922:
2918:
2913:
2909:
2904:
2900:
2896:
2894:
2882:
2873:p ≡ 3 mod 4
2859:p ≡ 1 mod 4
2852:
2847:
2828:
2822:
2807:
2803:
2801:
2726:
2722:
2624:
2559:
2555:
2551:
2527:
2523:
2499:
2495:
2491:
2489:
2483:
2475:
2471:
2469:
2387:
2312:
2310:
2239:
2235:
2231:
2229:
2223:
2215:
2212:discriminant
2207:
2203:
2201:
2052:
2048:
2046:
1978:
1974:
1970:
1968:
1903:
1901:
1842:
1841:ramifies in
1838:
1836:
1830:
1822:
1818:
1814:
1810:
1808:
1795:
1787:
1782:
1778:
1774:
1769:
1765:
1761:
1757:
1753:
1749:
1745:
1741:
1739:
1730:cubic fields
1723:
1714:
1710:
1696:
1691:
1683:
1679:
1675:
1671:
1666:
1662:
1658:
1654:
1574:
1570:
1565:
1561:
1536:
1532:
1528:
1524:
1518:
1514:
1510:
1506:
1502:
1496:
1492:
1488:
1482:
1478:
1474:
1470:
1464:
1460:
1456:
1452:
1447:
1443:
1439:
1433:
1429:
1425:
1419:
1415:
1411:
1405:
1400:
1394:
1390:
1386:
1380:
1376:
1372:
1366:
1362:
1358:
1238:
1234:
1230:
1225:
1221:
1217:
1213:
1208:
1204:
1200:
1194:
1190:
1186:
1184:
1179:
1175:
1169:
1165:
1161:
1157:
1153:
1149:
1144:
1140:
1136:
1131:
1127:
1123:
1117:
1113:
1109:
1105:
1104:| for every
1099:
1095:
1091:
1087:
1076:
1072:
1068:
1064:
1062:
1009:
919:
914:
910:
906:
901:
897:
893:
885:
881:
869:
865:
860:
856:
805:Galois group
792:
788:
786:
776:
772:
768:
763:
759:
758:= 1 (and so
754:
750:
746:
742:
738:
735:
730:
726:
725:= 1 (and so
721:
717:
713:
709:
705:
702:
698:
694:
689:
685:
680:
676:
674:
585:
576:
572:
567:
563:
556:
549:
541:
537:
533:
529:
525:
521:
520:ramifies in
517:
513:
509:
505:
501:
497:
493:
488:
484:
480:
475:
471:
469:
464:
459:
455:
451:
447:
442:
438:
434:
429:
425:
418:
413:
409:
405:
398:
394:
392:
386:
382:
377:
373:
369:
362:
355:
353:
267:
258:
251:
240:
236:
234:
144:
136:
132:
121:
114:
110:
106:
104:
94:
82:
74:
65:
61:
56:
52:
45:
43:prime ideals
38:
36:number field
31:
24:
22:Galois group
15:
3587:+ 1. Since
799:. Then the
446:= of this
270:dimensional
101:Definitions
18:mathematics
4526:Categories
4515:0956.11021
4456:PlanetMath
4431:Stein 2002
4385:References
3551:An example
3231:where the
3121:where the
2490:Any prime
2234:≡ 1 mod 4
2230:Any prime
1829:The prime
1773:is simply
1438:for given
1178:for every
876:under the
697:(and thus
588:ideal norm
538:unramified
536:is called
508:is called
479:is called
450:is called
433:for every
393:The field
4331:−
4319:⋯
4293:−
4187:⋯
4061:−
3637:conductor
3479:θ
3463:⊆
3434:∈
3406:conductor
3329:θ
3191:⋯
3070:⋯
2970:) modulo
2821:Prime in
2771:−
2765:≡
2683:−
2659:σ
2449:±
2424:≡
2348:±
2287:−
2165:−
2159:≡
2150:−
2129:−
2120:⋅
2093:−
2010:−
2004:≡
1719:preimages
1684:Artin map
1611:−
1607:σ
1586:σ
1548:σ
1321:
1274:
1268:→
1199:, called
1082:. By the
956:∏
823:
620:∑
300:∏
211:↪
199:↓
193:↓
174:↪
4477:(1999).
4373:See also
3392:= ... =
2926:so that
2313:switches
1981:, since
1809:Writing
1764:(i), so
1705:over an
1636:′
1108:, where
749:= 1 and
716:= 1 and
590:implies
510:ramified
4507:1697859
3635:), the
2814:Summary
2482:Primes
2222:Primes
855:on the
48:of the
4513:
4505:
4495:
3360:where
2607:
2370:
2236:splits
1802:, and
1777:, and
1122:, the
73:. The
3505:order
2496:inert
2055:, as
1678:into
1560:maps
884:over
874:orbit
773:inert
712:. If
492:over
463:over
263:field
261:is a
34:of a
27:of a
4493:ISBN
4237:and
3650:/(2)
3642:For
3583:) =
3535:and
2806:and
2651:and
2474:and
2204:only
2051:and
1817:(i)/
1798:has
1756:and
1744:(i)/
1713:and
1473:and
1174:is
1010:and
905:and
135:and
120:and
105:Let
4511:Zbl
4103:two
4075:mod
3998:= (
3864:= (
3715:mod
3378:to
3011:as
2899:of
2781:mod
2698:is
2560:not
2524:not
2498:in
2440:mod
2214:of
2175:mod
2020:mod
1833:= 2
1701:or
1569:to
1469:is
1399:is
1318:Gal
1271:Gal
1203:of
1126:of
1090:|/|
1075:in
880:of
868:in
820:Gal
775:in
771:is
741:in
708:in
675:If
540:at
512:at
483:of
454:of
368:by
131:of
89:of
16:In
4528::
4509:.
4503:MR
4501:.
4491:.
4481:.
4453:.
4433:,
4414:.
4264:13
4132:13
4079:13
3541:no
3524:.
3382:.
3276::
2879:1
2868:1
2865:1
2841:2
2810:.
2725:/7
2478:.
2419:13
2257:13
2199:.
1845::
1786:=
1760:=
1752:=
1721:.
1694:.
1578:,
1575:j'
1403:.
1182:.
1176:ef
1065:ef
909:=
896:=
779:.
684:=
557:pO
467:.
417:=
397:=
390:.
356:pO
265:.
4517:.
4459:.
4420:.
4357:.
4354:]
4351:i
4348:[
4344:Z
4340:)
4337:i
4334:3
4328:2
4325:(
4322:=
4316:=
4313:]
4310:i
4307:[
4303:Z
4299:)
4296:5
4290:i
4287:(
4284:+
4281:]
4278:i
4275:[
4271:Z
4267:)
4261:(
4258:=
4253:2
4249:Q
4222:]
4219:i
4216:[
4212:Z
4208:)
4205:i
4202:3
4199:+
4196:2
4193:(
4190:=
4184:=
4181:]
4178:i
4175:[
4171:Z
4167:)
4164:5
4161:+
4158:i
4155:(
4152:+
4149:]
4146:i
4143:[
4139:Z
4135:)
4129:(
4126:=
4121:1
4117:Q
4086:.
4082:)
4072:(
4067:)
4064:5
4058:X
4055:(
4052:)
4049:5
4046:+
4043:X
4040:(
4037:=
4034:1
4031:+
4026:2
4022:X
4008:P
4004:p
4000:p
3996:P
3979:.
3976:]
3973:i
3970:[
3966:Z
3962:7
3959:=
3956:]
3953:i
3950:[
3946:Z
3942:)
3939:1
3936:+
3931:2
3927:i
3923:(
3920:+
3917:]
3914:i
3911:[
3907:Z
3903:)
3900:7
3897:(
3894:=
3891:Q
3878:X
3874:P
3870:p
3866:p
3862:P
3845:.
3842:]
3839:i
3836:[
3832:Z
3828:)
3825:i
3822:+
3819:1
3816:(
3813:=
3810:]
3807:i
3804:[
3800:Z
3796:)
3793:1
3790:+
3787:i
3784:(
3781:+
3778:]
3775:i
3772:[
3768:Z
3764:)
3761:2
3758:(
3755:=
3752:Q
3726:.
3722:)
3719:2
3712:(
3705:2
3701:)
3697:1
3694:+
3691:X
3688:(
3685:=
3682:1
3679:+
3674:2
3670:X
3656:X
3652:Z
3648:Z
3644:P
3623:i
3615:(
3613:Q
3599:i
3591:[
3589:Z
3585:X
3581:X
3579:(
3577:H
3563:i
3545:P
3537:P
3533:K
3531:/
3529:L
3521:L
3517:O
3512:K
3508:O
3488:;
3485:}
3482:]
3476:[
3471:K
3467:O
3458:L
3454:O
3450:y
3447::
3442:L
3438:O
3431:y
3428:{
3414:K
3410:O
3398:n
3394:e
3390:1
3387:e
3380:K
3375:j
3371:h
3366:j
3362:h
3345:,
3340:L
3336:O
3332:)
3326:(
3321:j
3317:h
3313:+
3308:L
3304:O
3300:P
3297:=
3292:j
3288:Q
3273:j
3269:Q
3264:j
3260:h
3255:j
3251:Q
3246:L
3242:O
3237:j
3233:Q
3216:,
3209:n
3205:e
3199:n
3195:Q
3184:1
3180:e
3174:1
3170:Q
3166:=
3161:L
3157:O
3153:P
3140:P
3136:P
3132:F
3127:j
3123:h
3106:,
3099:n
3095:e
3090:)
3086:X
3083:(
3078:n
3074:h
3063:1
3059:e
3054:)
3050:X
3047:(
3042:1
3038:h
3034:=
3031:)
3028:X
3025:(
3022:h
3009:F
3005:X
3003:(
3001:h
2997:P
2995:/
2992:K
2988:O
2984:F
2980:X
2978:(
2976:h
2972:P
2968:X
2966:(
2964:H
2959:K
2955:O
2951:K
2947:X
2945:(
2943:H
2932:K
2928:L
2923:L
2919:O
2914:L
2910:O
2905:K
2901:O
2897:P
2883:G
2853:G
2848:G
2829:Z
2823:Z
2808:b
2804:a
2785:7
2777:i
2774:b
2768:a
2760:7
2756:)
2752:i
2749:b
2746:+
2743:a
2740:(
2727:Z
2723:Z
2709:i
2706:2
2686:i
2680:1
2677:=
2674:)
2671:i
2668:+
2665:1
2662:(
2639:i
2636:+
2633:1
2610:,
2602:L
2598:O
2594:)
2591:7
2588:(
2584:/
2578:L
2574:O
2556:p
2552:G
2538:i
2530:[
2528:Z
2510:i
2502:[
2500:Z
2492:p
2484:p
2476:b
2472:a
2455:i
2452:3
2444:2
2436:i
2433:b
2430:+
2427:a
2415:)
2411:i
2408:b
2405:+
2402:a
2399:(
2373:,
2365:L
2361:O
2357:)
2354:i
2351:3
2345:2
2342:(
2338:/
2332:L
2328:O
2296:)
2293:i
2290:3
2284:2
2281:(
2278:)
2275:i
2272:3
2269:+
2266:2
2263:(
2260:=
2240:Z
2232:p
2224:p
2216:Z
2208:Z
2187:i
2184:+
2179:1
2171:i
2168:b
2162:a
2156:i
2153:b
2147:a
2144:+
2141:i
2138:b
2135:)
2132:i
2126:1
2123:(
2117:)
2114:i
2111:+
2108:1
2105:(
2102:=
2099:i
2096:b
2090:a
2087:+
2084:i
2081:b
2078:2
2075:=
2072:i
2069:b
2066:+
2063:a
2053:b
2049:a
2032:i
2029:+
2024:1
2016:i
2013:b
2007:a
2001:i
1998:b
1995:+
1992:a
1979:G
1975:Z
1971:G
1952:L
1948:O
1944:)
1941:i
1938:+
1935:1
1932:(
1928:/
1922:L
1918:O
1904:e
1885:2
1881:)
1877:i
1874:+
1871:1
1868:(
1865:=
1862:)
1859:2
1856:(
1843:Z
1839:Z
1831:p
1823:G
1819:Q
1815:Q
1811:G
1796:Z
1788:Z
1783:L
1779:O
1775:Z
1770:K
1766:O
1762:Q
1758:L
1754:Q
1750:K
1746:Q
1742:Q
1692:K
1680:G
1676:K
1672:K
1667:j
1663:P
1659:P
1655:G
1633:j
1628:P
1623:D
1619:=
1614:1
1599:j
1595:P
1590:D
1571:P
1566:j
1562:P
1537:j
1533:P
1529:G
1525:G
1519:j
1515:P
1511:D
1507:j
1503:G
1497:j
1493:P
1489:D
1483:j
1479:P
1475:I
1471:f
1465:j
1461:P
1457:D
1453:F
1451:/
1448:j
1444:F
1440:j
1434:j
1430:P
1426:I
1424:/
1420:j
1416:P
1412:D
1401:e
1395:j
1391:P
1387:I
1381:j
1377:P
1373:I
1371:/
1367:j
1363:P
1359:D
1345:)
1342:F
1338:/
1332:j
1328:F
1324:(
1298:)
1295:F
1291:/
1285:j
1281:F
1277:(
1261:j
1257:P
1252:D
1239:j
1235:P
1231:I
1226:j
1222:F
1218:K
1216:/
1214:L
1209:j
1205:P
1195:j
1191:P
1187:I
1180:j
1170:j
1166:P
1162:D
1158:G
1154:K
1152:/
1150:L
1145:j
1141:P
1137:G
1132:j
1128:P
1118:j
1114:P
1110:D
1106:j
1100:j
1096:P
1092:D
1088:G
1079:L
1077:O
1073:p
1069:g
1048:.
1045:g
1042:f
1039:e
1036:=
1033:]
1030:K
1027::
1024:L
1021:[
1006:.
992:e
987:)
981:j
977:P
971:g
966:1
963:=
960:j
951:(
946:=
941:L
937:O
933:p
920:j
915:j
911:e
907:e
902:j
898:f
894:f
886:K
882:L
870:L
866:p
861:j
857:P
840:)
837:K
833:/
829:L
826:(
817:=
814:G
793:K
791:/
789:L
777:L
769:p
764:1
760:f
755:1
751:e
747:g
743:L
736:p
731:1
727:e
722:1
718:f
714:g
710:L
703:p
699:g
695:j
690:j
686:e
681:j
677:f
660:.
655:j
651:f
645:j
641:e
635:g
630:1
627:=
624:j
616:=
613:]
610:K
607::
604:L
601:[
577:K
575:/
573:L
568:j
564:F
559:L
555:/
552:L
550:O
542:p
534:K
532:/
530:L
526:L
522:L
518:p
514:p
506:K
504:/
502:L
498:j
494:p
489:j
485:P
476:j
472:e
465:p
460:j
456:P
443:j
439:f
435:j
430:j
426:P
424:/
421:L
419:O
414:j
410:F
406:p
404:/
401:K
399:O
395:F
387:j
383:e
378:j
374:P
370:p
365:L
363:O
358:L
335:j
331:e
325:j
321:P
315:g
310:1
307:=
304:j
296:=
291:L
287:O
283:p
259:p
257:/
254:K
252:O
243:K
241:O
237:p
216:L
206:K
184:L
180:O
167:K
163:O
145:Z
137:L
133:K
124:L
122:O
117:K
115:O
111:K
109:/
107:L
95:G
66:L
62:O
57:K
53:O
46:P
39:K
32:L
25:G
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.