38:
668:
400:
1901:
663:{\displaystyle \Delta _{K}=\det \left({\begin{array}{cccc}\sigma _{1}(b_{1})&\sigma _{1}(b_{2})&\cdots &\sigma _{1}(b_{n})\\\sigma _{2}(b_{1})&\ddots &&\vdots \\\vdots &&\ddots &\vdots \\\sigma _{n}(b_{1})&\cdots &\cdots &\sigma _{n}(b_{n})\end{array}}\right)^{2}.}
1840:
961:
1188:
2363:
1675:
835:
1417:
3732:"Extrait d'une lettre de M. C. Hermite à M. Borchardt sur le nombre limité d'irrationalités auxquelles se réduisent les racines des équations à coefficients entiers complexes d'un degré et d'un discriminant donnés"
2797:
1926:
first stated
Minkowski's theorem in 1882, though the first proof was given by Hermann Minkowski in 1891. In the same year, Minkowski published his bound on the discriminant. Near the end of the nineteenth century,
2725:
1626:
2543:
1038:
2913:
3028:
824:
2963:
1463:'s original example of a number field whose ring of integers does not possess a power basis. An integral basis is given by {1, α, α(α + 1)/2} and the discriminant of
2390:
1221:
2438:. An extension is unramified if, and only if, the discriminant is the unit ideal. The Minkowski bound above shows that there are no non-trivial unramified extensions of
1891:. Again, this follows from the Minkowski bound together with Hermite's theorem (that there are only finitely many algebraic number fields with prescribed discriminant).
2802:
There is also a lower bound that holds in all degrees, not just asymptotically: For totally real fields, the root discriminant is > 14, with 1229 exceptions.
3851:
2245:
1835:{\displaystyle |\Delta _{K}|^{1/2}\geq {\frac {n^{n}}{n!}}\left({\frac {\pi }{4}}\right)^{r_{2}}\geq {\frac {n^{n}}{n!}}\left({\frac {\pi }{4}}\right)^{n/2}.}
956:{\displaystyle \Delta _{K}=\left\{{\begin{array}{ll}d&{\text{if }}d\equiv 1{\pmod {4}}\\4d&{\text{if }}d\equiv 2,3{\pmod {4}}.\\\end{array}}\right.}
2548:
The relation between relative discriminants in a tower of fields shows that the root discriminant does not change in an unramified extension.
1341:
4260:
4218:
4124:
4085:
4029:
3984:
3947:
3913:
3142:
2741:
2672:
1918:
Hermite's theorem predates the general definition of the discriminant with
Charles Hermite publishing a proof of it in 1857. In 1877,
4170:
4059:
1978:. The relative discriminant is defined in a fashion similar to the absolute discriminant, but must take into account that ideals in
1266:
3345:
1571:
4116:
2480:
4320:
3967:; Diaz y Diaz, Francisco; Olivier, Michel (2002), "A Survey of Discriminant Counting", in Fieker, Claus; Kohel, David R. (eds.),
1907:
showed that every number field possesses an integral basis, allowing him to define the discriminant of an arbitrary number field.
135:
3044:
Due to its appearance in this volume, the discriminant also appears in the functional equation of the
Dedekind zeta function of
3498:
1183:{\displaystyle \Delta _{K_{n}}=(-1)^{\varphi (n)/2}{\frac {n^{\varphi (n)}}{\displaystyle \prod _{p|n}p^{\varphi (n)/(p-1)}}}}
162:
states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an
3705:
3341:
3096:
2732:
3628:
1915:, was given by Dedekind in 1871. At this point, he already knew the relationship between the discriminant and ramification.
1869:
1470:
Repeated discriminants: the discriminant of a quadratic field uniquely identifies it, but this is not true, in general, for
155:
3964:
3931:
3809:
3731:
2881:
147:
3774:
2447:
3685:
2972:
1644:
1224:
3278:
Theorem 1.3 (Dedekind). For a number field K, a prime p ramifies in K if and only if p divides the integer discZ(OK)
4021:
84:
3049:
31:
1437:
788:
37:
4163:
Algebraic Number Theory, Proceedings of an instructional conference at the
University of Sussex, Brighton, 1965
3556:
1471:
968:
2927:
104:
3969:
Algorithmic Number Theory, Proceedings, 5th
International Syposium, ANTS-V, University of Sydney, July 2002
4188:
3068:
1528:
139:
131:
100:
3270:
2966:
130:
The discriminant is one of the most basic invariants of a number field, and occurs in several important
4193:
Algorithmic number theory. Proceedings, 8th
International Symposium, ANTS-VIII, Banff, Canada, May 2008
4158:
4009:
3871:
3565:
1928:
1635:
1241:
365:
303:
2371:
3779:
3736:
2451:
1919:
782:
743:
197:
111:
of the) algebraic number field. More specifically, it is proportional to the squared volume of the
4255:, Graduate Texts in Mathematics, vol. 83 (2 nd ed.), Berlin, New York: Springer-Verlag,
4187:
Voight, John (2008), "Enumeration of totally real number fields of bounded root discriminant", in
4232:
4196:
3905:
3753:
3664:
3581:
1877:
be a positive integer. There are only finitely many (up to isomorphisms) algebraic number fields
1517:
1441:
1317:
1197:
112:
4195:, Lecture Notes in Computer Science, vol. 5011, Berlin: Springer-Verlag, pp. 268–281,
4102:
426:
3706:"Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Congruenzen"
4256:
4214:
4166:
4150:
4120:
4081:
4055:
4025:
3990:
3980:
3971:, Lecture Notes in Computer Science, vol. 2369, Berlin: Springer-Verlag, pp. 80–94,
3943:
3909:
3846:
3805:
3770:
3148:
3138:
2811:
2405:
2358:{\displaystyle \Delta _{K/F}={\mathcal {N}}_{L/F}\left({\Delta _{K/L}}\right)\Delta _{L/F}^{}}
1923:
331:
2844: = 3·5·7·11·19 produces fields of arbitrarily large degree with root discriminant 2
4274:
4240:
4206:
4154:
4138:
4065:
4013:
3972:
3897:
3879:
3860:
3829:
3821:
3788:
3745:
3701:
3681:
3648:
3640:
3589:
3573:
3506:
3156:
1904:
1460:
976:
774:
350:
270:
116:
108:
4270:
4228:
4180:
4134:
4095:
4039:
4002:
3957:
3923:
3660:
4278:
4266:
4244:
4224:
4176:
4142:
4130:
4091:
4069:
4051:
4035:
3998:
3953:
3939:
3919:
3883:
3864:
3833:
3792:
3727:
3656:
3652:
3593:
3160:
3084:
3064:
2225:
2185:
237:
186:
159:
3569:
4292:
3554:
Martinet, Jacques (1978). "Tours de corps de classes et estimations de discriminants".
3137:, Encyclopaedia of Mathematical Sciences, vol. 49 (Second ed.), p. 130,
747:
324:
290:
124:
4314:
3757:
3668:
3585:
3130:
990:
4236:
3810:"Ueber die positiven quadratischen Formen und über kettenbruchähnliche Algorithmen"
3072:
1262:
967:
An integer that occurs as the discriminant of a quadratic number field is called a
163:
120:
4296:
3874:(1897), "Über eine neue Eigenschaft der Diskriminanten algebraischer Zahlkörper",
4210:
17:
1478:
1475:
675:
354:
92:
79:
is 49 = 7. Accordingly, the volume of the fundamental domain is 7 and
204:, and like the absolute discriminant it indicates which primes are ramified in
3825:
3511:
3493:
2393:
2181:
1481:
of discriminant 3969. They are obtained by adjoining a root of the polynomial
1445:
1423:
which is exactly the definition of the discriminant of the minimal polynomial.
857:
687:
3994:
3749:
3152:
4080:, Springer Monographs in Mathematics (3 ed.), Berlin: Springer-Verlag,
3976:
328:
30:"Brill's theorem" redirects here. For the result in algebraic geometry, see
3876:
Proceedings of the First
International Congress of Mathematicians, Zürich
3710:
Abhandlungen der Königlichen
Gesellschaft der Wissenschaften zu Göttingen
2869:(0,1) < 82.2, improving upon earlier bounds of Martinet.
1911:
The definition of the discriminant of a general algebraic number field,
1412:{\displaystyle \prod _{1\leq i<j\leq n}(\alpha _{i}-\alpha _{j})^{2}}
3644:
3577:
2792:{\displaystyle \alpha (\rho ,\sigma )\geq 215.3^{\rho }44.7^{\sigma }.}
27:
Measures the size of the ring of integers of the algebraic number field
3494:"Tamely ramified towers and discriminant bounds for number fields. II"
1900:
2720:{\displaystyle \alpha (\rho ,\sigma )\geq 60.8^{\rho }22.3^{\sigma }}
1931:
obtained his theorem on the residue of the discriminant modulo four.
4050:, Encycl. Math. Sci., vol. 62 (2nd printing of 1st ed.),
212:. It is a generalization of the absolute discriminant allowing for
4201:
3938:, Graduate Texts in Mathematics, vol. 138, Berlin, New York:
3775:"Grundzüge einer arithmetischen Theorie der algebraischen Grössen"
1899:
36:
2446:
may have unramified extensions: for example, for any field with
1939:
The discriminant defined above is sometimes referred to as the
1621:{\displaystyle \Delta _{K}\equiv 0{\text{ or }}1{\pmod {4}}.}
2538:{\displaystyle \operatorname {rd} _{K}=|\Delta _{K}|^{1/n}.}
2377:
2273:
4020:, Cambridge Studies in Advanced Mathematics, vol. 27,
950:
3083:. This provides a relation to the Artin conductors of the
3048:, and hence in the analytic class number formula, and the
1865:| > 1 (this follows directly from the Minkowski bound).
1240:
Power bases: In the case where the ring of integers has a
41:
A fundamental domain of the ring of integers of the field
3687:
1269:
of α. To see this, one can choose the integral basis of
3115:
2428:
if, and only if, it divides the relative discriminant Δ
2975:
2930:
2884:
2826:). For example, the infinite class field tower over
2744:
2675:
2483:
2374:
2248:
1678:
1574:
1344:
1200:
1119:
1041:
838:
791:
403:
2908:{\displaystyle K\otimes _{\mathbf {Q} }\mathbf {R} }
1227:, and the product in the denominator is over primes
61: + 1. This fundamental domain sits inside
4078:
Elementary and analytic theory of algebraic numbers
3022:
2957:
2907:
2791:
2719:
2537:
2384:
2357:
1987:may not be principal and that there may not be an
1834:
1620:
1411:
1215:
1182:
955:
818:
662:
107:that, loosely speaking, measures the size of the (
3936:A Course in Computational Algebraic Number Theory
3340:Dedekind's supplement X of the second edition of
3023:{\displaystyle 2^{-r_{2}}{\sqrt {|\Delta _{K}|}}}
2810:On the other hand, the existence of an infinite
417:
3456:
3454:
2172:.) Alternatively, the relative discriminant of
3477:
4112:Grundlehren der mathematischen Wissenschaften
3487:
3485:
3432:
8:
4110:
3444:All facts in this paragraph can be found in
3336:
3334:
1316:. Then, the matrix in the definition is the
819:{\displaystyle K=\mathbf {Q} ({\sqrt {d}})}
3526:
3524:
3522:
3445:
3420:
3408:
3245:
3208:
3185:
2915:, the volume of the fundamental domain of
2239:the relative discriminants are related by
2071:) be the square of the determinant of the
1474:number fields. For example, there are two
244:generated by the absolute discriminant of
4200:
3852:Comptes rendus de l'Académie des sciences
3510:
3492:Hajir, Farshid; Maire, Christian (2002).
3396:
3013:
3007:
2998:
2996:
2988:
2980:
2974:
2948:
2942:
2933:
2931:
2929:
2900:
2893:
2892:
2883:
2780:
2770:
2743:
2711:
2701:
2674:
2522:
2518:
2513:
2506:
2497:
2488:
2482:
2376:
2375:
2373:
2337:
2328:
2324:
2305:
2301:
2296:
2282:
2278:
2272:
2271:
2257:
2253:
2247:
1922:determined the sign of the discriminant.
1819:
1815:
1801:
1780:
1774:
1763:
1758:
1744:
1723:
1717:
1704:
1700:
1695:
1688:
1679:
1677:
1599:
1591:
1579:
1573:
1403:
1393:
1380:
1349:
1343:
1199:
1155:
1142:
1128:
1124:
1104:
1098:
1088:
1075:
1051:
1046:
1040:
1020:th cyclotomic field. The discriminant of
927:
907:
879:
865:
856:
843:
837:
806:
798:
790:
651:
634:
621:
596:
583:
537:
524:
507:
494:
474:
461:
446:
433:
425:
408:
402:
3473:
3460:
3361:
3349:
3326:
3314:
3302:
3290:
3257:
3196:
2861:towers, Hajir and Maire have shown that
2404:The relative discriminant regulates the
2218:generated by the absolute discriminant Δ
1520:of the discriminant is (−1) where
4153:(1967), "Local class field theory", in
3372:
3125:
3123:
3108:
2958:{\displaystyle {\sqrt {|\Delta _{K}|}}}
2814:can give upper bounds on the values of
2454:is a non-trivial unramified extension.
735:, so the square of the discriminant of
166:, and the subject of current research.
3902:Elements of the history of mathematics
3849:(1891b), "Théorèmes d'arithmétiques",
3542:
2102:). Then, the relative discriminant of
983: > 2 be an integer, let ζ
686:can be used. Specifically, define the
260:be an algebraic number field, and let
3607:
3384:
3220:
3173:
3116:Cohen, Diaz y Diaz & Olivier 2002
2857:(0,1) < 296.276. Using
2564:, not both 0, and a positive integer
2154:. (i.e. bases with the property that
7:
3530:
3232:
3135:Introduction to Modern Number Theory
2146:} varies over all integral bases of
1436:(α) be the number field obtained by
3271:"Discriminants and ramified primes"
3037:is the number of complex places of
2969:is used and the volume obtained is
2556:Given nonnegative rational numbers
1607:
935:
928:
887:
880:
739:is the determinant of this matrix.
196:of number fields. The latter is an
3004:
2939:
2503:
2321:
2298:
2250:
1685:
1576:
1043:
840:
405:
25:
4253:Introduction to Cyclotomic Fields
1961:of an extension of number fields
1332:(α), whose determinant squared is
2901:
2894:
1662:the number of complex places of
799:
2865:(1,0) < 954.3 and
1600:
228:, the relative discriminant of
4076:Narkiewicz, Władysław (2004),
3346:Vorlesungen über Zahlentheorie
3342:Peter Gustav Lejeune Dirichlet
3097:conductor-discriminant formula
3014:
2999:
2949:
2934:
2760:
2748:
2733:generalized Riemann hypothesis
2691:
2679:
2514:
2498:
2385:{\displaystyle {\mathcal {N}}}
2350:
2338:
2110:is the ideal generated by the
2015:} be the set of embeddings of
1696:
1680:
1611:
1601:
1459: − 8. This is
1400:
1373:
1210:
1204:
1172:
1160:
1152:
1146:
1129:
1114:
1108:
1085:
1079:
1072:
1062:
939:
929:
891:
881:
813:
803:
640:
627:
602:
589:
543:
530:
513:
500:
480:
467:
452:
439:
319:} be the set of embeddings of
1:
4251:Washington, Lawrence (1997),
3627:Brill, Alexander von (1877),
3133:; Panchishkin, A. A. (2007),
3055:The relative discriminant of
2204:, the relative discriminant Δ
1244:, that is, can be written as
148:analytic class number formula
4211:10.1007/978-3-540-79456-1_18
2873:Relation to other quantities
2631:complex embeddings, and let
2408:data of the field extension
765:is defined in the same way.
3908:. Berlin: Springer-Verlag.
3476:or Proposition III.2.15 of
2735:implies the stronger bound
1947:to distinguish it from the
1216:{\displaystyle \varphi (n)}
785:, then the discriminant of
200:in the ring of integers of
181:to distinguish it from the
4337:
4189:van der Poorten, Alfred J.
4165:, London: Academic Press,
4107:Algebraische Zahlentheorie
4022:Cambridge University Press
3478:Fröhlich & Taylor 1993
2474:is defined by the formula
2214:is the principal ideal of
2023:which are the identity on
173:can be referred to as the
29:
4191:; Stein, Andreas (eds.),
4115:. Vol. 322. Berlin:
3826:10.1515/crll.1891.107.278
3629:"Ueber die Discriminante"
1870:Hermite–Minkowski theorem
119:, and it regulates which
3750:10.1515/crll.1857.53.182
3557:Inventiones Mathematicae
3301:Proposition III.2.14 of
1225:Euler's totient function
969:fundamental discriminant
690:to be the matrix whose (
4321:Algebraic number theory
4298:Algebraic Number Theory
4048:Algebraic Number Theory
4018:Algebraic number theory
3977:10.1007/3-540-45455-1_7
3512:10.1023/A:1017537415688
3087:of the Galois group of
2965:(sometimes a different
2806:Asymptotic upper bounds
2552:Asymptotic lower bounds
1969:, which is an ideal in
1560:Stickelberger's theorem
775:Quadratic number fields
742:The discriminant of an
49:by adjoining a root of
4111:
3472:Corollary III.2.10 of
3256:Corollary III.2.12 of
3069:regular representation
3024:
2959:
2909:
2793:
2721:
2609:) be the infimum of rd
2539:
2450:greater than one, its
2386:
2359:
1908:
1836:
1622:
1413:
1257:, the discriminant of
1217:
1184:
957:
820:
728:). This matrix equals
664:
140:Dedekind zeta function
101:algebraic number field
88:
75:. The discriminant of
4046:Koch, Helmut (1997),
3872:Stickelberger, Ludwig
3633:Mathematische Annalen
3050:Brauer–Siegel theorem
3025:
2960:
2910:
2794:
2722:
2643:) = liminf
2627:real embeddings and 2
2540:
2442:. Fields larger than
2387:
2360:
1949:relative discriminant
1935:Relative discriminant
1903:
1837:
1623:
1414:
1218:
1185:
958:
821:
665:
183:relative discriminant
175:absolute discriminant
134:formulas such as the
40:
32:Brill–Noether theorem
3878:, pp. 182–193,
3690:(2 ed.), Vieweg
3325:Theorem III.2.16 of
3313:Theorem III.2.17 of
3172:Definition 5.1.2 of
2973:
2928:
2882:
2742:
2673:
2568:such that the pair (
2481:
2372:
2246:
1929:Ludwig Stickelberger
1676:
1572:
1455: − 2
1342:
1242:power integral basis
1198:
1039:
836:
789:
401:
169:The discriminant of
57: − 2
3570:1978InMat..44...65M
3499:J. Symbolic Comput.
3184:Proposition 2.7 of
2878:When embedded into
2623:number fields with
2619:ranges over degree
2452:Hilbert class field
2354:
1920:Alexander von Brill
1849:Minkowski's theorem
1451: −
783:square-free integer
302:(i.e. a basis as a
136:functional equation
53: −
4159:Fröhlich, Albrecht
4151:Serre, Jean-Pierre
4010:Fröhlich, Albrecht
3847:Minkowski, Hermann
3806:Minkowski, Hermann
3771:Kronecker, Leopold
3645:10.1007/BF01442468
3578:10.1007/bf01389902
3533:, pp. 181–182
3433:Stickelberger 1897
3289:Exercise I.2.7 of
3020:
2955:
2905:
2789:
2717:
2535:
2382:
2355:
2320:
1909:
1832:
1618:
1409:
1372:
1318:Vandermonde matrix
1267:minimal polynomial
1213:
1180:
1177:
1137:
953:
948:
816:
674:Equivalently, the
660:
645:
332:ring homomorphisms
216:to be bigger than
113:fundamental domain
89:
4262:978-0-387-94762-4
4220:978-3-540-79455-4
4155:Cassels, J. W. S.
4126:978-3-540-65399-8
4087:978-3-540-21902-6
4031:978-0-521-43834-6
3986:978-3-540-43863-2
3949:978-3-540-55640-4
3915:978-3-540-64767-6
3898:Bourbaki, Nicolas
3891:Secondary sources
3702:Dedekind, Richard
3682:Dedekind, Richard
3448:, pp. 59, 81
3144:978-3-540-20364-3
3018:
2953:
2812:class field tower
2464:root discriminant
2458:Root discriminant
2392:denotes relative
1924:Leopold Kronecker
1809:
1794:
1752:
1737:
1647:of the extension
1636:Minkowski's bound
1594:
1527:is the number of
1345:
1178:
1120:
977:Cyclotomic fields
910:
868:
811:
394:). Symbolically,
18:Root-discriminant
16:(Redirected from
4328:
4307:
4306:
4305:
4281:
4247:
4204:
4183:
4146:
4114:
4103:Neukirch, Jürgen
4098:
4072:
4042:
4005:
3960:
3927:
3904:. Translated by
3886:
3867:
3842:
3841:
3840:
3820:(107): 278–297,
3814:Crelle's Journal
3801:
3800:
3799:
3780:Crelle's Journal
3766:
3765:
3764:
3737:Crelle's Journal
3728:Hermite, Charles
3723:
3722:
3721:
3697:
3696:
3695:
3677:
3676:
3675:
3610:
3604:
3598:
3597:
3551:
3545:
3540:
3534:
3528:
3517:
3516:
3514:
3489:
3480:
3470:
3464:
3458:
3449:
3442:
3436:
3430:
3424:
3418:
3412:
3406:
3400:
3394:
3388:
3382:
3376:
3370:
3364:
3359:
3353:
3338:
3329:
3323:
3317:
3311:
3305:
3299:
3293:
3287:
3281:
3280:
3275:
3266:
3260:
3254:
3248:
3242:
3236:
3230:
3224:
3218:
3212:
3206:
3200:
3194:
3188:
3182:
3176:
3170:
3164:
3163:
3127:
3118:
3113:
3029:
3027:
3026:
3021:
3019:
3017:
3012:
3011:
3002:
2997:
2995:
2994:
2993:
2992:
2964:
2962:
2961:
2956:
2954:
2952:
2947:
2946:
2937:
2932:
2914:
2912:
2911:
2906:
2904:
2899:
2898:
2897:
2852:
2851:
2839:
2838:
2798:
2796:
2795:
2790:
2785:
2784:
2775:
2774:
2726:
2724:
2723:
2718:
2716:
2715:
2706:
2705:
2544:
2542:
2541:
2536:
2531:
2530:
2526:
2517:
2511:
2510:
2501:
2493:
2492:
2416:. A prime ideal
2391:
2389:
2388:
2383:
2381:
2380:
2364:
2362:
2361:
2356:
2353:
2336:
2332:
2319:
2315:
2314:
2313:
2309:
2291:
2290:
2286:
2277:
2276:
2266:
2265:
2261:
2043:is any basis of
1943:discriminant of
1905:Richard Dedekind
1841:
1839:
1838:
1833:
1828:
1827:
1823:
1814:
1810:
1802:
1795:
1793:
1785:
1784:
1775:
1770:
1769:
1768:
1767:
1757:
1753:
1745:
1738:
1736:
1728:
1727:
1718:
1713:
1712:
1708:
1699:
1693:
1692:
1683:
1627:
1625:
1624:
1619:
1614:
1595:
1592:
1584:
1583:
1502:
1491:
1461:Richard Dedekind
1418:
1416:
1415:
1410:
1408:
1407:
1398:
1397:
1385:
1384:
1371:
1292: = α,
1285: = 1,
1261:is equal to the
1222:
1220:
1219:
1214:
1189:
1187:
1186:
1181:
1179:
1176:
1175:
1159:
1136:
1132:
1118:
1117:
1099:
1097:
1096:
1092:
1058:
1057:
1056:
1055:
995:th root of unity
962:
960:
959:
954:
952:
949:
942:
911:
908:
894:
869:
866:
848:
847:
825:
823:
822:
817:
812:
807:
802:
669:
667:
666:
661:
656:
655:
650:
646:
639:
638:
626:
625:
601:
600:
588:
587:
565:
552:
542:
541:
529:
528:
512:
511:
499:
498:
479:
478:
466:
465:
451:
450:
438:
437:
413:
412:
271:ring of integers
220:; in fact, when
117:ring of integers
109:ring of integers
21:
4336:
4335:
4331:
4330:
4329:
4327:
4326:
4325:
4311:
4310:
4303:
4301:
4293:Milne, James S.
4291:
4288:
4286:Further reading
4263:
4250:
4221:
4186:
4173:
4149:
4127:
4117:Springer-Verlag
4101:
4088:
4075:
4062:
4052:Springer-Verlag
4045:
4032:
4008:
3987:
3963:
3950:
3940:Springer-Verlag
3930:
3916:
3896:
3893:
3870:
3845:
3838:
3836:
3804:
3797:
3795:
3769:
3762:
3760:
3744:(53): 182–192,
3726:
3719:
3717:
3700:
3693:
3691:
3680:
3673:
3671:
3626:
3623:
3621:Primary sources
3618:
3613:
3606:Section 4.4 of
3605:
3601:
3553:
3552:
3548:
3541:
3537:
3529:
3520:
3491:
3490:
3483:
3471:
3467:
3459:
3452:
3446:Narkiewicz 2004
3443:
3439:
3431:
3427:
3421:Minkowski 1891b
3419:
3415:
3409:Minkowski 1891a
3407:
3403:
3395:
3391:
3383:
3379:
3371:
3367:
3360:
3356:
3339:
3332:
3324:
3320:
3312:
3308:
3300:
3296:
3288:
3284:
3273:
3269:Conrad, Keith.
3268:
3267:
3263:
3255:
3251:
3246:Washington 1997
3243:
3239:
3231:
3227:
3223:, Theorem 6.4.6
3219:
3215:
3209:Narkiewicz 2004
3207:
3203:
3195:
3191:
3186:Washington 1997
3183:
3179:
3171:
3167:
3145:
3129:
3128:
3121:
3114:
3110:
3106:
3065:Artin conductor
3036:
3003:
2984:
2976:
2971:
2970:
2938:
2926:
2925:
2923:
2888:
2880:
2879:
2875:
2859:tamely ramified
2847:
2845:
2833:
2831:
2808:
2776:
2766:
2740:
2739:
2707:
2697:
2671:
2670:
2657:
2649:
2614:
2600:
2576:) = (
2554:
2512:
2502:
2484:
2479:
2478:
2460:
2437:
2402:
2370:
2369:
2297:
2292:
2270:
2249:
2244:
2243:
2226:tower of fields
2223:
2213:
2166:
2159:
2145:
2136:
2129:
2120:
2101:
2092:
2070:
2061:
2042:
2033:
2014:
2008:
2004:
1995:
1986:
1977:
1960:
1937:
1898:
1886:
1864:
1797:
1796:
1786:
1776:
1759:
1740:
1739:
1729:
1719:
1694:
1684:
1674:
1673:
1661:
1575:
1570:
1569:
1555:
1546:if and only if
1526:
1514:Brill's theorem
1510:
1503:, respectively.
1493:
1482:
1399:
1389:
1376:
1340:
1339:
1331:
1325:
1320:associated to α
1311:
1298:
1291:
1284:
1277:
1252:
1196:
1195:
1138:
1100:
1071:
1047:
1042:
1037:
1036:
1028:
1015:
1005:
988:
947:
946:
905:
896:
895:
863:
852:
839:
834:
833:
787:
786:
771:
764:
755:
727:
719:
710:
673:
644:
643:
630:
617:
615:
610:
605:
592:
579:
576:
575:
570:
564:
558:
557:
551:
546:
533:
520:
517:
516:
503:
490:
488:
483:
470:
457:
455:
442:
429:
421:
420:
404:
399:
398:
393:
384:
325:complex numbers
318:
312:
301:
288:
279:
268:
254:
238:principal ideal
103:is a numerical
70:
35:
28:
23:
22:
15:
12:
11:
5:
4334:
4332:
4324:
4323:
4313:
4312:
4309:
4308:
4287:
4284:
4283:
4282:
4261:
4248:
4219:
4184:
4171:
4147:
4125:
4099:
4086:
4073:
4060:
4043:
4030:
4014:Taylor, Martin
4006:
3985:
3961:
3948:
3928:
3914:
3892:
3889:
3888:
3887:
3868:
3843:
3802:
3767:
3724:
3698:
3678:
3622:
3619:
3617:
3614:
3612:
3611:
3599:
3546:
3535:
3518:
3481:
3465:
3450:
3437:
3425:
3413:
3401:
3397:Kronecker 1882
3389:
3377:
3365:
3354:
3330:
3318:
3306:
3294:
3282:
3261:
3249:
3237:
3225:
3213:
3201:
3189:
3177:
3165:
3143:
3119:
3107:
3105:
3102:
3101:
3100:
3053:
3042:
3034:
3016:
3010:
3006:
3001:
2991:
2987:
2983:
2979:
2951:
2945:
2941:
2936:
2919:
2903:
2896:
2891:
2887:
2874:
2871:
2853:≈ 296.276, so
2807:
2804:
2800:
2799:
2788:
2783:
2779:
2773:
2769:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2729:
2728:
2714:
2710:
2704:
2700:
2696:
2693:
2690:
2687:
2684:
2681:
2678:
2653:
2644:
2610:
2596:
2588: × 2
2553:
2550:
2546:
2545:
2534:
2529:
2525:
2521:
2516:
2509:
2505:
2500:
2496:
2491:
2487:
2459:
2456:
2429:
2401:
2398:
2379:
2366:
2365:
2352:
2349:
2346:
2343:
2340:
2335:
2331:
2327:
2323:
2318:
2312:
2308:
2304:
2300:
2295:
2289:
2285:
2281:
2275:
2269:
2264:
2260:
2256:
2252:
2219:
2205:
2164:
2157:
2141:
2134:
2125:
2118:
2097:
2088:
2079:matrix whose (
2066:
2059:
2038:
2031:
2010:
2006:
2000:
1991:
1982:
1973:
1952:
1936:
1933:
1897:
1894:
1893:
1892:
1882:
1866:
1860:
1845:
1844:
1843:
1842:
1831:
1826:
1822:
1818:
1813:
1808:
1805:
1800:
1792:
1789:
1783:
1779:
1773:
1766:
1762:
1756:
1751:
1748:
1743:
1735:
1732:
1726:
1722:
1716:
1711:
1707:
1703:
1698:
1691:
1687:
1682:
1668:
1667:
1659:
1631:
1630:
1629:
1628:
1617:
1613:
1610:
1606:
1603:
1598:
1593: or
1590:
1587:
1582:
1578:
1564:
1563:
1557:
1551:
1536:
1529:complex places
1524:
1509:
1506:
1505:
1504:
1476:non-isomorphic
1468:
1467:is −503.
1425:
1424:
1421:
1420:
1419:
1406:
1402:
1396:
1392:
1388:
1383:
1379:
1375:
1370:
1367:
1364:
1361:
1358:
1355:
1352:
1348:
1334:
1333:
1327:
1326: = σ
1321:
1307:
1296:
1289:
1282:
1273:
1248:
1237:
1236:
1212:
1209:
1206:
1203:
1192:
1191:
1190:
1174:
1171:
1168:
1165:
1162:
1158:
1154:
1151:
1148:
1145:
1141:
1135:
1131:
1127:
1123:
1116:
1113:
1110:
1107:
1103:
1095:
1091:
1087:
1084:
1081:
1078:
1074:
1070:
1067:
1064:
1061:
1054:
1050:
1045:
1031:
1030:
1024:
1011:
1001:
984:
973:
972:
965:
964:
963:
951:
945:
941:
938:
934:
931:
926:
923:
920:
917:
914:
906:
904:
901:
898:
897:
893:
890:
886:
883:
878:
875:
872:
864:
862:
859:
858:
855:
851:
846:
842:
828:
827:
815:
810:
805:
801:
797:
794:
770:
767:
760:
753:
748:integral basis
723:
715:
702:
671:
670:
659:
654:
649:
642:
637:
633:
629:
624:
620:
616:
614:
611:
609:
606:
604:
599:
595:
591:
586:
582:
578:
577:
574:
571:
569:
566:
563:
560:
559:
556:
553:
550:
547:
545:
540:
536:
532:
527:
523:
519:
518:
515:
510:
506:
502:
497:
493:
489:
487:
484:
482:
477:
473:
469:
464:
460:
456:
454:
449:
445:
441:
436:
432:
428:
427:
424:
419:
416:
411:
407:
389:
380:
314:
310:
297:
291:integral basis
284:
277:
264:
253:
250:
66:
45:obtained from
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4333:
4322:
4319:
4318:
4316:
4300:
4299:
4294:
4290:
4289:
4285:
4280:
4276:
4272:
4268:
4264:
4258:
4254:
4249:
4246:
4242:
4238:
4234:
4230:
4226:
4222:
4216:
4212:
4208:
4203:
4198:
4194:
4190:
4185:
4182:
4178:
4174:
4172:0-12-163251-2
4168:
4164:
4160:
4156:
4152:
4148:
4144:
4140:
4136:
4132:
4128:
4122:
4118:
4113:
4108:
4104:
4100:
4097:
4093:
4089:
4083:
4079:
4074:
4071:
4067:
4063:
4061:3-540-63003-1
4057:
4053:
4049:
4044:
4041:
4037:
4033:
4027:
4023:
4019:
4015:
4011:
4007:
4004:
4000:
3996:
3992:
3988:
3982:
3978:
3974:
3970:
3966:
3962:
3959:
3955:
3951:
3945:
3941:
3937:
3933:
3929:
3925:
3921:
3917:
3911:
3907:
3906:Meldrum, John
3903:
3899:
3895:
3894:
3890:
3885:
3881:
3877:
3873:
3869:
3866:
3862:
3858:
3854:
3853:
3848:
3844:
3835:
3831:
3827:
3823:
3819:
3815:
3811:
3807:
3803:
3794:
3790:
3786:
3782:
3781:
3776:
3772:
3768:
3759:
3755:
3751:
3747:
3743:
3739:
3738:
3733:
3729:
3725:
3715:
3711:
3707:
3703:
3699:
3689:
3688:
3683:
3679:
3670:
3666:
3662:
3658:
3654:
3650:
3646:
3642:
3638:
3634:
3630:
3625:
3624:
3620:
3615:
3609:
3603:
3600:
3595:
3591:
3587:
3583:
3579:
3575:
3571:
3567:
3563:
3560:(in French).
3559:
3558:
3550:
3547:
3544:
3539:
3536:
3532:
3527:
3525:
3523:
3519:
3513:
3508:
3504:
3501:
3500:
3495:
3488:
3486:
3482:
3479:
3475:
3474:Neukirch 1999
3469:
3466:
3462:
3461:Neukirch 1999
3457:
3455:
3451:
3447:
3441:
3438:
3434:
3429:
3426:
3422:
3417:
3414:
3410:
3405:
3402:
3398:
3393:
3390:
3386:
3381:
3378:
3374:
3369:
3366:
3363:
3362:Bourbaki 1994
3358:
3355:
3351:
3350:Dedekind 1871
3347:
3343:
3337:
3335:
3331:
3328:
3327:Neukirch 1999
3322:
3319:
3316:
3315:Neukirch 1999
3310:
3307:
3304:
3303:Neukirch 1999
3298:
3295:
3292:
3291:Neukirch 1999
3286:
3283:
3279:
3272:
3265:
3262:
3259:
3258:Neukirch 1999
3253:
3250:
3247:
3244:Lemma 2.2 of
3241:
3238:
3234:
3229:
3226:
3222:
3217:
3214:
3210:
3205:
3202:
3198:
3197:Dedekind 1878
3193:
3190:
3187:
3181:
3178:
3175:
3169:
3166:
3162:
3158:
3154:
3150:
3146:
3140:
3136:
3132:
3131:Manin, Yu. I.
3126:
3124:
3120:
3117:
3112:
3109:
3103:
3098:
3095:, called the
3094:
3090:
3086:
3082:
3078:
3074:
3070:
3066:
3062:
3058:
3054:
3051:
3047:
3043:
3040:
3033:
3008:
2989:
2985:
2981:
2977:
2968:
2943:
2922:
2918:
2889:
2885:
2877:
2876:
2872:
2870:
2868:
2864:
2860:
2856:
2850:
2843:
2837:
2829:
2825:
2821:
2817:
2813:
2805:
2803:
2786:
2781:
2777:
2771:
2767:
2763:
2757:
2754:
2751:
2745:
2738:
2737:
2736:
2734:
2712:
2708:
2702:
2698:
2694:
2688:
2685:
2682:
2676:
2669:
2668:
2667:
2665:
2661:
2656:
2652:
2647:
2642:
2638:
2634:
2630:
2626:
2622:
2618:
2613:
2608:
2604:
2599:
2595:
2591:
2587:
2583:
2579:
2575:
2571:
2567:
2563:
2559:
2551:
2549:
2532:
2527:
2523:
2519:
2507:
2494:
2489:
2485:
2477:
2476:
2475:
2473:
2470:number field
2469:
2465:
2457:
2455:
2453:
2449:
2445:
2441:
2436:
2432:
2427:
2423:
2419:
2415:
2411:
2407:
2399:
2397:
2395:
2347:
2344:
2341:
2333:
2329:
2325:
2316:
2310:
2306:
2302:
2293:
2287:
2283:
2279:
2267:
2262:
2258:
2254:
2242:
2241:
2240:
2238:
2234:
2230:
2227:
2224: . In a
2222:
2217:
2212:
2208:
2203:
2199:
2195:
2191:
2187:
2183:
2179:
2175:
2171:
2167:
2161: ∈
2160:
2153:
2149:
2144:
2140:
2133:
2128:
2124:
2117:
2113:
2109:
2105:
2100:
2096:
2091:
2086:
2082:
2078:
2074:
2069:
2065:
2058:
2054:
2050:
2046:
2041:
2037:
2030:
2026:
2022:
2018:
2013:
2003:
1999:
1994:
1990:
1985:
1981:
1976:
1972:
1968:
1964:
1959:
1955:
1950:
1946:
1942:
1934:
1932:
1930:
1925:
1921:
1916:
1914:
1906:
1902:
1895:
1890:
1885:
1880:
1876:
1872:
1871:
1867:
1863:
1858:
1854:
1850:
1847:
1846:
1829:
1824:
1820:
1816:
1811:
1806:
1803:
1798:
1790:
1787:
1781:
1777:
1771:
1764:
1760:
1754:
1749:
1746:
1741:
1733:
1730:
1724:
1720:
1714:
1709:
1705:
1701:
1689:
1672:
1671:
1670:
1669:
1665:
1658:
1654:
1650:
1646:
1642:
1638:
1637:
1633:
1632:
1615:
1608:
1604:
1596:
1588:
1585:
1580:
1568:
1567:
1566:
1565:
1561:
1558:
1554:
1549:
1545:
1541:
1537:
1534:
1530:
1523:
1519:
1515:
1512:
1511:
1508:Basic results
1507:
1500:
1496:
1489:
1485:
1480:
1477:
1473:
1472:higher-degree
1469:
1466:
1462:
1458:
1454:
1450:
1447:
1443:
1439:
1435:
1431:
1427:
1426:
1422:
1404:
1394:
1390:
1386:
1381:
1377:
1368:
1365:
1362:
1359:
1356:
1353:
1350:
1346:
1338:
1337:
1336:
1335:
1330:
1324:
1319:
1315:
1312: =
1310:
1306:
1302:
1299: =
1295:
1288:
1281:
1276:
1272:
1268:
1264:
1260:
1256:
1251:
1247:
1243:
1239:
1238:
1234:
1230:
1226:
1207:
1201:
1193:
1169:
1166:
1163:
1156:
1149:
1143:
1139:
1133:
1125:
1121:
1111:
1105:
1101:
1093:
1089:
1082:
1076:
1068:
1065:
1059:
1052:
1048:
1035:
1034:
1033:
1032:
1027:
1023:
1019:
1014:
1009:
1004:
1000:
996:
994:
987:
982:
978:
975:
974:
970:
966:
943:
936:
932:
924:
921:
918:
915:
912:
902:
899:
888:
884:
876:
873:
870:
860:
853:
849:
844:
832:
831:
830:
829:
808:
795:
792:
784:
780:
776:
773:
772:
768:
766:
763:
759:
752:
749:
745:
740:
738:
734:
731:
726:
722:
718:
714:
709:
705:
701:
697:
693:
689:
685:
681:
677:
657:
652:
647:
635:
631:
622:
618:
612:
607:
597:
593:
584:
580:
572:
567:
561:
554:
548:
538:
534:
525:
521:
508:
504:
495:
491:
485:
475:
471:
462:
458:
447:
443:
434:
430:
422:
414:
409:
397:
396:
395:
392:
388:
383:
378:
374:
370:
367:
364:
360:
356:
352:
348:
344:
340:
337: →
336:
333:
330:
326:
322:
317:
309:), and let {σ
308:
306:
300:
296:
292:
287:
283:
276:
272:
267:
263:
259:
251:
249:
247:
243:
239:
235:
231:
227:
224: =
223:
219:
215:
211:
207:
203:
199:
195:
191:
188:
184:
180:
176:
172:
167:
165:
161:
157:
153:
149:
145:
141:
137:
133:
128:
126:
122:
118:
114:
110:
106:
102:
98:
94:
86:
82:
78:
74:
69:
65: ⊗
64:
60:
56:
52:
48:
44:
39:
33:
19:
4302:, retrieved
4297:
4252:
4192:
4162:
4106:
4077:
4047:
4017:
3968:
3965:Cohen, Henri
3935:
3932:Cohen, Henri
3901:
3875:
3856:
3850:
3837:, retrieved
3817:
3813:
3796:, retrieved
3784:
3778:
3761:, retrieved
3741:
3735:
3718:, retrieved
3713:
3709:
3692:, retrieved
3686:
3672:, retrieved
3639:(1): 87–89,
3636:
3632:
3602:
3561:
3555:
3549:
3538:
3502:
3497:
3468:
3440:
3428:
3416:
3404:
3392:
3380:
3373:Hermite 1857
3368:
3357:
3321:
3309:
3297:
3285:
3277:
3264:
3252:
3240:
3235:, p. 11
3228:
3216:
3204:
3192:
3180:
3168:
3134:
3111:
3092:
3088:
3080:
3076:
3073:Galois group
3060:
3056:
3045:
3038:
3031:
2920:
2916:
2866:
2862:
2858:
2854:
2848:
2841:
2835:
2827:
2823:
2819:
2815:
2809:
2801:
2730:
2663:
2659:
2654:
2650:
2645:
2640:
2636:
2632:
2628:
2624:
2620:
2616:
2611:
2606:
2602:
2597:
2593:
2589:
2585:
2581:
2577:
2573:
2569:
2565:
2561:
2557:
2555:
2547:
2471:
2467:
2466:of a degree
2463:
2461:
2448:class number
2443:
2439:
2434:
2430:
2425:
2424:ramifies in
2421:
2417:
2413:
2409:
2406:ramification
2403:
2400:Ramification
2367:
2236:
2232:
2228:
2220:
2215:
2210:
2206:
2201:
2197:
2193:
2189:
2177:
2173:
2169:
2162:
2155:
2151:
2147:
2142:
2138:
2131:
2126:
2122:
2115:
2111:
2107:
2103:
2098:
2094:
2089:
2087:)-entry is σ
2084:
2080:
2076:
2072:
2067:
2063:
2056:
2052:
2048:
2044:
2039:
2035:
2028:
2024:
2020:
2016:
2011:
2001:
1997:
1992:
1988:
1983:
1979:
1974:
1970:
1966:
1962:
1957:
1953:
1948:
1944:
1940:
1938:
1917:
1912:
1910:
1888:
1883:
1878:
1874:
1868:
1861:
1856:
1852:
1848:
1663:
1656:
1652:
1648:
1640:
1634:
1559:
1552:
1547:
1543:
1542:ramifies in
1539:
1532:
1521:
1513:
1498:
1494:
1487:
1483:
1479:cubic fields
1464:
1456:
1452:
1448:
1433:
1429:
1328:
1322:
1313:
1308:
1304:
1300:
1293:
1286:
1279:
1274:
1270:
1263:discriminant
1258:
1254:
1249:
1245:
1232:
1228:
1025:
1021:
1017:
1012:
1007:
1002:
998:
992:
985:
980:
778:
761:
757:
750:
741:
736:
732:
729:
724:
720:
716:
712:
707:
703:
699:
695:
691:
683:
679:
672:
390:
386:
381:
379:)-entry is σ
376:
372:
368:
362:
358:
346:
343:discriminant
342:
338:
334:
320:
315:
304:
298:
294:
285:
281:
274:
265:
261:
257:
255:
245:
241:
233:
229:
225:
221:
217:
213:
209:
205:
201:
193:
189:
182:
178:
174:
170:
168:
164:open problem
151:
143:
129:
97:discriminant
96:
90:
80:
76:
72:
67:
62:
58:
54:
50:
46:
42:
3859:: 209–212,
3543:Voight 2008
3505:: 415–423.
3199:, pp. 30–31
1643:denote the
1029:is given by
698:)-entry is
355:determinant
93:mathematics
4304:2008-08-20
4279:0966.11047
4245:1205.11125
4143:0956.11021
4070:0819.11044
3884:29.0172.03
3865:23.0214.01
3839:2009-08-20
3834:23.0212.01
3798:2009-08-20
3793:14.0038.02
3763:2009-08-20
3720:2009-08-20
3694:2009-08-05
3674:2009-08-22
3653:09.0059.02
3616:References
3608:Serre 1967
3594:0369.12007
3385:Brill 1877
3221:Cohen 1993
3174:Cohen 1993
3161:1079.11002
3085:characters
1501:− 35
1497:− 21
1486:− 21
1446:polynomial
997:, and let
991:primitive
746:in K with
688:trace form
252:Definition
146:, and the
4202:0802.0194
3995:0302-9743
3808:(1891a),
3787:: 1–122,
3758:120694650
3669:120947279
3586:122278145
3564:: 65–73.
3531:Koch 1997
3233:Koch 1997
3153:0938-0396
3005:Δ
2982:−
2940:Δ
2890:⊗
2782:σ
2772:ρ
2764:≥
2758:σ
2752:ρ
2746:α
2713:σ
2703:ρ
2695:≥
2689:σ
2683:ρ
2677:α
2666:). Then
2504:Δ
2322:Δ
2299:Δ
2251:Δ
2186:different
1996:basis of
1859:, then |Δ
1804:π
1772:≥
1747:π
1715:≥
1686:Δ
1586:≡
1577:Δ
1550:divides Δ
1444:α of the
1438:adjoining
1391:α
1387:−
1378:α
1366:≤
1354:≤
1347:∏
1231:dividing
1202:φ
1167:−
1144:φ
1122:∏
1106:φ
1077:φ
1066:−
1044:Δ
1016:) be the
916:≡
874:≡
841:Δ
619:σ
613:⋯
608:⋯
581:σ
573:⋮
568:⋱
562:⋮
555:⋮
549:⋱
522:σ
492:σ
486:⋯
459:σ
431:σ
406:Δ
329:injective
323:into the
187:extension
156:A theorem
105:invariant
4315:Category
4295:(1998),
4237:30036220
4161:(eds.),
4105:(1999).
4016:(1993),
3934:(1993),
3900:(1994).
3773:(1882),
3730:(1857),
3704:(1878),
3684:(1871),
3463:, §III.2
3030:, where
2731:and the
2648:→∞
2584:) is in
2168:for all
2009:, ..., σ
2005:. Let {σ
1941:absolute
1538:A prime
909:if
867:if
769:Examples
313:, ..., σ
132:analytic
125:ramified
85:ramified
83:is only
4271:1421575
4229:2467853
4181:0220701
4135:1697859
4096:2078267
4040:1215934
4003:2041075
3958:1228206
3924:1290116
3661:1509928
3566:Bibcode
3211:, p. 64
3071:of the
3067:of the
3063:is the
2967:measure
2846:√
2840:) with
2832:√
2822:,
2662:,
2639:,
2605:,
2196:. When
2184:of the
2180:is the
2137:, ...,
2121:, ...,
2062:, ...,
2034:, ...,
1896:History
1887:| <
1881:with |Δ
1855:is not
1556: .
1303:, ...,
1265:of the
756:, ...,
371:whose (
357:of the
353:of the
349:is the
341:). The
307:-module
280:, ...,
269:be its
236:is the
160:Hermite
138:of the
115:of the
4277:
4269:
4259:
4243:
4235:
4227:
4217:
4179:
4169:
4141:
4133:
4123:
4094:
4084:
4068:
4058:
4038:
4028:
4001:
3993:
3983:
3956:
3946:
3922:
3912:
3882:
3863:
3832:
3791:
3756:
3667:
3659:
3651:
3592:
3584:
3159:
3151:
3141:
2592:, let
2368:where
2130:) as {
2051:, let
1873:: Let
1666:, then
1645:degree
1639:: Let
1516:: The
1278:to be
1194:where
979:: let
777:: let
366:matrix
351:square
327:(i.e.
289:be an
273:. Let
185:of an
121:primes
99:of an
95:, the
71:
4233:S2CID
4197:arXiv
3754:S2CID
3665:S2CID
3582:S2CID
3274:(PDF)
3104:Notes
2768:215.3
2047:over
2027:. If
2019:into
1851:: If
989:be a
781:be a
744:order
678:from
676:trace
198:ideal
87:at 7.
4257:ISBN
4215:ISBN
4167:ISBN
4121:ISBN
4082:ISBN
4056:ISBN
4026:ISBN
3991:ISSN
3981:ISBN
3944:ISBN
3910:ISBN
3818:1891
3742:1857
3149:ISSN
3139:ISBN
2778:44.7
2709:22.3
2699:60.8
2560:and
2462:The
2394:norm
2182:norm
1655:and
1518:sign
1490:+ 28
1442:root
1428:Let
1360:<
256:Let
150:for
123:are
4275:Zbl
4241:Zbl
4207:doi
4139:Zbl
4066:Zbl
3973:doi
3880:JFM
3861:JFM
3857:112
3830:JFM
3822:doi
3789:JFM
3746:doi
3716:(1)
3649:JFM
3641:doi
3590:Zbl
3574:doi
3507:doi
3344:'s
3157:Zbl
3075:of
2924:is
2615:as
2420:of
2188:of
2075:by
1605:mod
1531:of
1492:or
1223:is
933:mod
885:mod
682:to
418:det
361:by
345:of
293:of
240:of
177:of
158:of
142:of
91:In
4317::
4273:,
4267:MR
4265:,
4239:,
4231:,
4225:MR
4223:,
4213:,
4205:,
4177:MR
4175:,
4157:;
4137:.
4131:MR
4129:.
4119:.
4109:.
4092:MR
4090:,
4064:,
4054:,
4036:MR
4034:,
4024:,
4012:;
3999:MR
3997:,
3989:,
3979:,
3954:MR
3952:,
3942:,
3920:MR
3918:.
3855:,
3828:,
3816:,
3812:,
3785:92
3783:,
3777:,
3752:,
3740:,
3734:,
3714:23
3712:,
3708:,
3663:,
3657:MR
3655:,
3647:,
3637:12
3635:,
3631:,
3588:.
3580:.
3572:.
3562:44
3521:^
3503:33
3496:.
3484:^
3453:^
3333:^
3276:.
3155:,
3147:,
3122:^
3041:).
2582:σn
2578:ρn
2572:,2
2486:rd
2396:.
2200:=
1440:a
1432:=
1253:=
1010:(ζ
1006:=
826:is
700:Tr
248:.
154:.
127:.
4209::
4199::
4145:.
3975::
3926:.
3824::
3748::
3643::
3596:.
3576::
3568::
3515:.
3509::
3435:.
3423:.
3411:.
3399:.
3387:.
3375:.
3352:)
3348:(
3099:.
3093:L
3091:/
3089:K
3081:L
3079:/
3077:K
3061:L
3059:/
3057:K
3052:.
3046:K
3039:K
3035:2
3032:r
3015:|
3009:K
3000:|
2990:2
2986:r
2978:2
2950:|
2944:K
2935:|
2921:K
2917:O
2902:R
2895:Q
2886:K
2867:α
2863:α
2855:α
2849:m
2842:m
2836:m
2834:-
2830:(
2828:Q
2824:σ
2820:ρ
2818:(
2816:α
2787:.
2761:)
2755:,
2749:(
2727:,
2692:)
2686:,
2680:(
2664:σ
2660:ρ
2658:(
2655:n
2651:α
2646:n
2641:σ
2637:ρ
2635:(
2633:α
2629:s
2625:r
2621:n
2617:K
2612:K
2607:σ
2603:ρ
2601:(
2598:n
2594:α
2590:Z
2586:Z
2580:,
2574:s
2570:r
2566:n
2562:σ
2558:ρ
2533:.
2528:n
2524:/
2520:1
2515:|
2508:K
2499:|
2495:=
2490:K
2472:K
2468:n
2444:Q
2440:Q
2435:L
2433:/
2431:K
2426:K
2422:L
2418:p
2414:L
2412:/
2410:K
2378:N
2351:]
2348:L
2345::
2342:K
2339:[
2334:F
2330:/
2326:L
2317:)
2311:L
2307:/
2303:K
2294:(
2288:F
2284:/
2280:L
2274:N
2268:=
2263:F
2259:/
2255:K
2237:F
2235:/
2233:L
2231:/
2229:K
2221:K
2216:Z
2211:Q
2209:/
2207:K
2202:Q
2198:L
2194:L
2192:/
2190:K
2178:L
2176:/
2174:K
2170:i
2165:K
2163:O
2158:i
2156:b
2152:L
2150:/
2148:K
2143:n
2139:b
2135:1
2132:b
2127:n
2123:b
2119:1
2116:b
2114:(
2112:d
2108:L
2106:/
2104:K
2099:j
2095:b
2093:(
2090:i
2085:j
2083:,
2081:i
2077:n
2073:n
2068:n
2064:b
2060:1
2057:b
2055:(
2053:d
2049:L
2045:K
2040:n
2036:b
2032:1
2029:b
2025:L
2021:C
2017:K
2012:n
2007:1
2002:K
1998:O
1993:L
1989:O
1984:L
1980:O
1975:L
1971:O
1967:L
1965:/
1963:K
1958:L
1956:/
1954:K
1951:Δ
1945:K
1913:K
1889:N
1884:K
1879:K
1875:N
1862:K
1857:Q
1853:K
1830:.
1825:2
1821:/
1817:n
1812:)
1807:4
1799:(
1791:!
1788:n
1782:n
1778:n
1765:2
1761:r
1755:)
1750:4
1742:(
1734:!
1731:n
1725:n
1721:n
1710:2
1706:/
1702:1
1697:|
1690:K
1681:|
1664:K
1660:2
1657:r
1653:Q
1651:/
1649:K
1641:n
1616:.
1612:)
1609:4
1602:(
1597:1
1589:0
1581:K
1562::
1553:K
1548:p
1544:K
1540:p
1535:.
1533:K
1525:2
1522:r
1499:x
1495:x
1488:x
1484:x
1465:K
1457:x
1453:x
1449:x
1434:Q
1430:K
1405:2
1401:)
1395:j
1382:i
1374:(
1369:n
1363:j
1357:i
1351:1
1329:i
1323:i
1314:α
1309:n
1305:b
1301:α
1297:3
1294:b
1290:2
1287:b
1283:1
1280:b
1275:K
1271:O
1259:K
1255:Z
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