27:
657:
389:
1890:
652:{\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}.}
1829:
950:
1177:
2352:
1664:
824:
1406:
3721:"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"
2786:
1915:
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,
2714:
1615:
2532:
1027:
2902:
3017:
813:
2952:
1452:'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
2379:
1210:
2427:. 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
1880:. Again, this follows from the Minkowski bound together with Hermite's theorem (that there are only finitely many algebraic number fields with prescribed discriminant).
2791:
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.
3840:
2234:
1824:{\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}.}
945:{\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.}
2537:
The relation between relative discriminants in a tower of fields shows that the root discriminant does not change in an unramified extension.
1330:
4249:
4207:
4113:
4074:
4018:
3973:
3936:
3902:
3131:
2730:
2661:
1907:
Hermite's theorem predates the general definition of the discriminant with
Charles Hermite publishing a proof of it in 1857. In 1877,
4159:
4048:
1967:. The relative discriminant is defined in a fashion similar to the absolute discriminant, but must take into account that ideals in
1255:
3334:
1560:
4105:
2469:
4309:
3956:; Diaz y Diaz, Francisco; Olivier, Michel (2002), "A Survey of Discriminant Counting", in Fieker, Claus; Kohel, David R. (eds.),
1896:
showed that every number field possesses an integral basis, allowing him to define the discriminant of an arbitrary number field.
124:
3033:
Due to its appearance in this volume, the discriminant also appears in the functional equation of the
Dedekind zeta function of
3487:
1172:{\displaystyle \Delta _{K_{n}}=(-1)^{\varphi (n)/2}{\frac {n^{\varphi (n)}}{\displaystyle \prod _{p|n}p^{\varphi (n)/(p-1)}}}}
151:
states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an
3694:
3330:
3085:
2721:
3617:
1904:, was given by Dedekind in 1871. At this point, he already knew the relationship between the discriminant and ramification.
1858:
1459:
Repeated discriminants: the discriminant of a quadratic field uniquely identifies it, but this is not true, in general, for
144:
3953:
3920:
3798:
3720:
2870:
136:
3763:
2436:
3674:
2961:
1633:
1213:
3267:
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)
4010:
73:
3038:
20:
1426:
777:
26:
4152:
Algebraic Number Theory, Proceedings of an instructional conference at the
University of Sussex, Brighton, 1965
3545:
1460:
957:
2916:
93:
3958:
Algorithmic Number Theory, Proceedings, 5th
International Syposium, ANTS-V, University of Sydney, July 2002
4177:
3057:
1517:
128:
120:
89:
3259:
2955:
119:
The discriminant is one of the most basic invariants of a number field, and occurs in several important
4182:
Algorithmic number theory. Proceedings, 8th
International Symposium, ANTS-VIII, Banff, Canada, May 2008
4147:
3998:
3860:
3554:
1917:
1624:
1230:
354:
292:
2360:
3768:
3725:
2440:
1908:
771:
732:
186:
100:
of the) algebraic number field. More specifically, it is proportional to the squared volume of the
4244:, Graduate Texts in Mathematics, vol. 83 (2 nd ed.), Berlin, New York: Springer-Verlag,
4176:
Voight, John (2008), "Enumeration of totally real number fields of bounded root discriminant", in
4221:
4185:
3894:
3742:
3653:
3570:
1866:
be a positive integer. There are only finitely many (up to isomorphisms) algebraic number fields
1506:
1430:
1306:
1186:
101:
4184:, Lecture Notes in Computer Science, vol. 5011, Berlin: Springer-Verlag, pp. 268–281,
4091:
415:
3695:"Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Congruenzen"
4245:
4203:
4155:
4139:
4109:
4070:
4044:
4014:
3979:
3969:
3960:, Lecture Notes in Computer Science, vol. 2369, Berlin: Springer-Verlag, pp. 80–94,
3932:
3898:
3835:
3794:
3759:
3137:
3127:
2800:
2394:
2347:{\displaystyle \Delta _{K/F}={\mathcal {N}}_{L/F}\left({\Delta _{K/L}}\right)\Delta _{L/F}^{}}
1912:
320:
2833: = 3·5·7·11·19 produces fields of arbitrarily large degree with root discriminant 2
4263:
4229:
4195:
4143:
4127:
4054:
4002:
3961:
3886:
3868:
3849:
3818:
3810:
3777:
3734:
3690:
3670:
3637:
3629:
3578:
3562:
3495:
3145:
1893:
1449:
965:
763:
339:
259:
105:
97:
4259:
4217:
4169:
4123:
4084:
4028:
3991:
3946:
3912:
3649:
4267:
4255:
4233:
4213:
4165:
4131:
4119:
4080:
4058:
4040:
4024:
3987:
3942:
3928:
3908:
3872:
3853:
3822:
3781:
3716:
3645:
3641:
3582:
3149:
3073:
3053:
2214:
2174:
226:
175:
148:
3558:
4281:
3543:
Martinet, Jacques (1978). "Tours de corps de classes et estimations de discriminants".
3126:, Encyclopaedia of Mathematical Sciences, vol. 49 (Second ed.), p. 130,
736:
313:
279:
113:
4303:
3746:
3657:
3574:
3119:
979:
4225:
3799:"Ueber die positiven quadratischen Formen und über kettenbruchähnliche Algorithmen"
3061:
1251:
956:
An integer that occurs as the discriminant of a quadratic number field is called a
152:
109:
4285:
3863:(1897), "Über eine neue Eigenschaft der Diskriminanten algebraischer Zahlkörper",
4199:
1467:
1464:
664:
343:
81:
68:
is 49 = 7. Accordingly, the volume of the fundamental domain is 7 and
193:, and like the absolute discriminant it indicates which primes are ramified in
3814:
3500:
3482:
2382:
2170:
1470:
of discriminant 3969. They are obtained by adjoining a root of the polynomial
1434:
1412:
which is exactly the definition of the discriminant of the minimal polynomial.
846:
676:
3983:
3738:
3141:
4069:, Springer Monographs in Mathematics (3 ed.), Berlin: Springer-Verlag,
3965:
317:
19:"Brill's theorem" redirects here. For the result in algebraic geometry, see
3865:
Proceedings of the First
International Congress of Mathematicians, Zürich
3699:
Abhandlungen der Königlichen
Gesellschaft der Wissenschaften zu Göttingen
2858:(0,1) < 82.2, improving upon earlier bounds of Martinet.
1900:
The definition of the discriminant of a general algebraic number field,
1401:{\displaystyle \prod _{1\leq i<j\leq n}(\alpha _{i}-\alpha _{j})^{2}}
3633:
3566:
2781:{\displaystyle \alpha (\rho ,\sigma )\geq 215.3^{\rho }44.7^{\sigma }.}
16:
Measures the size of the ring of integers of the algebraic number field
3483:"Tamely ramified towers and discriminant bounds for number fields. II"
1889:
2709:{\displaystyle \alpha (\rho ,\sigma )\geq 60.8^{\rho }22.3^{\sigma }}
1920:
obtained his theorem on the residue of the discriminant modulo four.
4039:, Encycl. Math. Sci., vol. 62 (2nd printing of 1st ed.),
201:. It is a generalization of the absolute discriminant allowing for
4190:
3927:, Graduate Texts in Mathematics, vol. 138, Berlin, New York:
3764:"Grundzüge einer arithmetischen Theorie der algebraischen Grössen"
1888:
25:
2435:
may have unramified extensions: for example, for any field with
1928:
The discriminant defined above is sometimes referred to as the
1610:{\displaystyle \Delta _{K}\equiv 0{\text{ or }}1{\pmod {4}}.}
2527:{\displaystyle \operatorname {rd} _{K}=|\Delta _{K}|^{1/n}.}
2366:
2262:
4009:, Cambridge Studies in Advanced Mathematics, vol. 27,
939:
3072:. This provides a relation to the Artin conductors of the
3037:, and hence in the analytic class number formula, and the
1854:| > 1 (this follows directly from the Minkowski bound).
1229:
Power bases: In the case where the ring of integers has a
30:
A fundamental domain of the ring of integers of the field
3676:
1258:
of α. To see this, one can choose the integral basis of
3104:
2417:
if, and only if, it divides the relative discriminant Δ
2964:
2919:
2873:
2815:). For example, the infinite class field tower over
2733:
2664:
2472:
2363:
2237:
1667:
1563:
1333:
1189:
1108:
1030:
827:
780:
392:
2897:{\displaystyle K\otimes _{\mathbf {Q} }\mathbf {R} }
1216:, and the product in the denominator is over primes
50: + 1. This fundamental domain sits inside
4067:
Elementary and analytic theory of algebraic numbers
3011:
2946:
2896:
2780:
2708:
2526:
2373:
2346:
1976:may not be principal and that there may not be an
1823:
1609:
1400:
1204:
1171:
944:
807:
651:
96:that, loosely speaking, measures the size of the (
3925:A Course in Computational Algebraic Number Theory
3329:Dedekind's supplement X of the second edition of
3012:{\displaystyle 2^{-r_{2}}{\sqrt {|\Delta _{K}|}}}
2799:On the other hand, the existence of an infinite
406:
3445:
3443:
2161:.) Alternatively, the relative discriminant of
3466:
4101:Grundlehren der mathematischen Wissenschaften
3476:
3474:
3421:
8:
4099:
3433:All facts in this paragraph can be found in
3325:
3323:
1305:. Then, the matrix in the definition is the
808:{\displaystyle K=\mathbf {Q} ({\sqrt {d}})}
3515:
3513:
3511:
3434:
3409:
3397:
3234:
3197:
3174:
2904:, the volume of the fundamental domain of
2228:the relative discriminants are related by
2060:) be the square of the determinant of the
1463:number fields. For example, there are two
233:generated by the absolute discriminant of
4189:
3841:Comptes rendus de l'Académie des sciences
3499:
3481:Hajir, Farshid; Maire, Christian (2002).
3385:
3002:
2996:
2987:
2985:
2977:
2969:
2963:
2937:
2931:
2922:
2920:
2918:
2889:
2882:
2881:
2872:
2769:
2759:
2732:
2700:
2690:
2663:
2511:
2507:
2502:
2495:
2486:
2477:
2471:
2365:
2364:
2362:
2326:
2317:
2313:
2294:
2290:
2285:
2271:
2267:
2261:
2260:
2246:
2242:
2236:
1911:determined the sign of the discriminant.
1808:
1804:
1790:
1769:
1763:
1752:
1747:
1733:
1712:
1706:
1693:
1689:
1684:
1677:
1668:
1666:
1588:
1580:
1568:
1562:
1392:
1382:
1369:
1338:
1332:
1188:
1144:
1131:
1117:
1113:
1093:
1087:
1077:
1064:
1040:
1035:
1029:
1009:th cyclotomic field. The discriminant of
916:
896:
868:
854:
845:
832:
826:
795:
787:
779:
640:
623:
610:
585:
572:
526:
513:
496:
483:
463:
450:
435:
422:
414:
397:
391:
3462:
3449:
3350:
3338:
3315:
3303:
3291:
3279:
3246:
3185:
2850:towers, Hajir and Maire have shown that
2393:The relative discriminant regulates the
2207:generated by the absolute discriminant Δ
1509:of the discriminant is (−1) where
4142:(1967), "Local class field theory", in
3361:
3114:
3112:
3097:
2947:{\displaystyle {\sqrt {|\Delta _{K}|}}}
2803:can give upper bounds on the values of
2443:is a non-trivial unramified extension.
724:, so the square of the discriminant of
155:, and the subject of current research.
3891:Elements of the history of mathematics
3838:(1891b), "Théorèmes d'arithmétiques",
3531:
2091:). Then, the relative discriminant of
972: > 2 be an integer, let ζ
675:can be used. Specifically, define the
249:be an algebraic number field, and let
3596:
3373:
3209:
3162:
3105:Cohen, Diaz y Diaz & Olivier 2002
2846:(0,1) < 296.276. Using
2553:, not both 0, and a positive integer
2143:. (i.e. bases with the property that
7:
3519:
3221:
3124:Introduction to Modern Number Theory
2135:} varies over all integral bases of
1425:(α) be the number field obtained by
3260:"Discriminants and ramified primes"
3026:is the number of complex places of
2958:is used and the volume obtained is
2545:Given nonnegative rational numbers
1596:
924:
917:
876:
869:
728:is the determinant of this matrix.
185:of number fields. The latter is an
2993:
2928:
2492:
2310:
2287:
2239:
1674:
1565:
1032:
829:
394:
14:
4242:Introduction to Cyclotomic Fields
1950:of an extension of number fields
1321:(α), whose determinant squared is
2890:
2883:
1651:the number of complex places of
788:
2854:(1,0) < 954.3 and
1589:
217:, the relative discriminant of
4065:Narkiewicz, Władysław (2004),
3335:Vorlesungen über Zahlentheorie
3331:Peter Gustav Lejeune Dirichlet
3086:conductor-discriminant formula
3003:
2988:
2938:
2923:
2749:
2737:
2722:generalized Riemann hypothesis
2680:
2668:
2503:
2487:
2374:{\displaystyle {\mathcal {N}}}
2339:
2327:
2099:is the ideal generated by the
2004:} be the set of embeddings of
1685:
1669:
1600:
1590:
1448: − 8. This is
1389:
1362:
1199:
1193:
1161:
1149:
1141:
1135:
1118:
1103:
1097:
1074:
1068:
1061:
1051:
928:
918:
880:
870:
802:
792:
629:
616:
591:
578:
532:
519:
502:
489:
469:
456:
441:
428:
308:} be the set of embeddings of
1:
4240:Washington, Lawrence (1997),
3616:Brill, Alexander von (1877),
3122:; Panchishkin, A. A. (2007),
3044:The relative discriminant of
2193:, the relative discriminant Δ
1233:, that is, can be written as
137:analytic class number formula
4200:10.1007/978-3-540-79456-1_18
2862:Relation to other quantities
2620:complex embeddings, and let
2397:data of the field extension
754:is defined in the same way.
3897:. Berlin: Springer-Verlag.
3465:or Proposition III.2.15 of
2724:implies the stronger bound
1936:to distinguish it from the
1205:{\displaystyle \varphi (n)}
774:, then the discriminant of
189:in the ring of integers of
170:to distinguish it from the
4326:
4178:van der Poorten, Alfred J.
4154:, London: Academic Press,
4096:Algebraische Zahlentheorie
4011:Cambridge University Press
3467:Fröhlich & Taylor 1993
2463:is defined by the formula
2203:is the principal ideal of
2012:which are the identity on
162:can be referred to as the
18:
4180:; Stein, Andreas (eds.),
4104:. Vol. 322. Berlin:
3815:10.1515/crll.1891.107.278
3618:"Ueber die Discriminante"
1859:Hermite–Minkowski theorem
108:, and it regulates which
3739:10.1515/crll.1857.53.182
3546:Inventiones Mathematicae
3290:Proposition III.2.14 of
1214:Euler's totient function
958:fundamental discriminant
679:to be the matrix whose (
4310:Algebraic number theory
4287:Algebraic Number Theory
4037:Algebraic Number Theory
4007:Algebraic number theory
3966:10.1007/3-540-45455-1_7
3501:10.1023/A:1017537415688
3076:of the Galois group of
2954:(sometimes a different
2795:Asymptotic upper bounds
2541:Asymptotic lower bounds
1958:, which is an ideal in
1549:Stickelberger's theorem
764:Quadratic number fields
731:The discriminant of an
38:by adjoining a root of
4100:
3461:Corollary III.2.10 of
3245:Corollary III.2.12 of
3058:regular representation
3013:
2948:
2898:
2782:
2710:
2598:) be the infimum of rd
2528:
2439:greater than one, its
2375:
2348:
1897:
1825:
1611:
1402:
1246:, the discriminant of
1206:
1173:
946:
809:
717:). This matrix equals
653:
129:Dedekind zeta function
90:algebraic number field
77:
64:. The discriminant of
4035:Koch, Helmut (1997),
3861:Stickelberger, Ludwig
3622:Mathematische Annalen
3039:Brauer–Siegel theorem
3014:
2949:
2899:
2783:
2711:
2632:) = liminf
2616:real embeddings and 2
2529:
2431:. Fields larger than
2376:
2349:
1938:relative discriminant
1924:Relative discriminant
1892:
1826:
1612:
1403:
1207:
1174:
947:
810:
654:
172:relative discriminant
164:absolute discriminant
123:formulas such as the
29:
21:Brill–Noether theorem
3867:, pp. 182–193,
3679:(2 ed.), Vieweg
3314:Theorem III.2.16 of
3302:Theorem III.2.17 of
3161:Definition 5.1.2 of
2962:
2917:
2871:
2731:
2662:
2557:such that the pair (
2470:
2361:
2235:
1918:Ludwig Stickelberger
1665:
1561:
1444: − 2
1331:
1231:power integral basis
1187:
1028:
825:
778:
390:
158:The discriminant of
46: − 2
3559:1978InMat..44...65M
3488:J. Symbolic Comput.
3173:Proposition 2.7 of
2867:When embedded into
2612:number fields with
2608:ranges over degree
2441:Hilbert class field
2343:
1909:Alexander von Brill
1838:Minkowski's theorem
1440: −
772:square-free integer
291:(i.e. a basis as a
125:functional equation
42: −
4148:Fröhlich, Albrecht
4140:Serre, Jean-Pierre
3999:Fröhlich, Albrecht
3836:Minkowski, Hermann
3795:Minkowski, Hermann
3760:Kronecker, Leopold
3634:10.1007/BF01442468
3567:10.1007/bf01389902
3522:, pp. 181–182
3422:Stickelberger 1897
3278:Exercise I.2.7 of
3009:
2944:
2894:
2778:
2706:
2524:
2371:
2344:
2309:
1898:
1821:
1607:
1398:
1361:
1307:Vandermonde matrix
1256:minimal polynomial
1202:
1169:
1166:
1126:
942:
937:
805:
663:Equivalently, the
649:
634:
321:ring homomorphisms
205:to be bigger than
102:fundamental domain
78:
4251:978-0-387-94762-4
4209:978-3-540-79455-4
4144:Cassels, J. W. S.
4115:978-3-540-65399-8
4076:978-3-540-21902-6
4020:978-0-521-43834-6
3975:978-3-540-43863-2
3938:978-3-540-55640-4
3904:978-3-540-64767-6
3887:Bourbaki, Nicolas
3880:Secondary sources
3691:Dedekind, Richard
3671:Dedekind, Richard
3437:, pp. 59, 81
3133:978-3-540-20364-3
3007:
2942:
2801:class field tower
2453:root discriminant
2447:Root discriminant
2381:denotes relative
1913:Leopold Kronecker
1798:
1783:
1741:
1726:
1636:of the extension
1625:Minkowski's bound
1583:
1516:is the number of
1334:
1167:
1109:
966:Cyclotomic fields
899:
857:
800:
383:). Symbolically,
4317:
4296:
4295:
4294:
4270:
4236:
4193:
4172:
4135:
4103:
4092:Neukirch, Jürgen
4087:
4061:
4031:
3994:
3949:
3916:
3893:. Translated by
3875:
3856:
3831:
3830:
3829:
3809:(107): 278–297,
3803:Crelle's Journal
3790:
3789:
3788:
3769:Crelle's Journal
3755:
3754:
3753:
3726:Crelle's Journal
3717:Hermite, Charles
3712:
3711:
3710:
3686:
3685:
3684:
3666:
3665:
3664:
3599:
3593:
3587:
3586:
3540:
3534:
3529:
3523:
3517:
3506:
3505:
3503:
3478:
3469:
3459:
3453:
3447:
3438:
3431:
3425:
3419:
3413:
3407:
3401:
3395:
3389:
3383:
3377:
3371:
3365:
3359:
3353:
3348:
3342:
3327:
3318:
3312:
3306:
3300:
3294:
3288:
3282:
3276:
3270:
3269:
3264:
3255:
3249:
3243:
3237:
3231:
3225:
3219:
3213:
3207:
3201:
3195:
3189:
3183:
3177:
3171:
3165:
3159:
3153:
3152:
3116:
3107:
3102:
3018:
3016:
3015:
3010:
3008:
3006:
3001:
3000:
2991:
2986:
2984:
2983:
2982:
2981:
2953:
2951:
2950:
2945:
2943:
2941:
2936:
2935:
2926:
2921:
2903:
2901:
2900:
2895:
2893:
2888:
2887:
2886:
2841:
2840:
2828:
2827:
2787:
2785:
2784:
2779:
2774:
2773:
2764:
2763:
2715:
2713:
2712:
2707:
2705:
2704:
2695:
2694:
2533:
2531:
2530:
2525:
2520:
2519:
2515:
2506:
2500:
2499:
2490:
2482:
2481:
2405:. A prime ideal
2380:
2378:
2377:
2372:
2370:
2369:
2353:
2351:
2350:
2345:
2342:
2325:
2321:
2308:
2304:
2303:
2302:
2298:
2280:
2279:
2275:
2266:
2265:
2255:
2254:
2250:
2032:is any basis of
1932:discriminant of
1894:Richard Dedekind
1830:
1828:
1827:
1822:
1817:
1816:
1812:
1803:
1799:
1791:
1784:
1782:
1774:
1773:
1764:
1759:
1758:
1757:
1756:
1746:
1742:
1734:
1727:
1725:
1717:
1716:
1707:
1702:
1701:
1697:
1688:
1682:
1681:
1672:
1616:
1614:
1613:
1608:
1603:
1584:
1581:
1573:
1572:
1491:
1480:
1450:Richard Dedekind
1407:
1405:
1404:
1399:
1397:
1396:
1387:
1386:
1374:
1373:
1360:
1281: = α,
1274: = 1,
1250:is equal to the
1211:
1209:
1208:
1203:
1178:
1176:
1175:
1170:
1168:
1165:
1164:
1148:
1125:
1121:
1107:
1106:
1088:
1086:
1085:
1081:
1047:
1046:
1045:
1044:
984:th root of unity
951:
949:
948:
943:
941:
938:
931:
900:
897:
883:
858:
855:
837:
836:
814:
812:
811:
806:
801:
796:
791:
658:
656:
655:
650:
645:
644:
639:
635:
628:
627:
615:
614:
590:
589:
577:
576:
554:
541:
531:
530:
518:
517:
501:
500:
488:
487:
468:
467:
455:
454:
440:
439:
427:
426:
402:
401:
260:ring of integers
209:; in fact, when
106:ring of integers
98:ring of integers
4325:
4324:
4320:
4319:
4318:
4316:
4315:
4314:
4300:
4299:
4292:
4290:
4282:Milne, James S.
4280:
4277:
4275:Further reading
4252:
4239:
4210:
4175:
4162:
4138:
4116:
4106:Springer-Verlag
4090:
4077:
4064:
4051:
4041:Springer-Verlag
4034:
4021:
3997:
3976:
3952:
3939:
3929:Springer-Verlag
3919:
3905:
3885:
3882:
3859:
3834:
3827:
3825:
3793:
3786:
3784:
3758:
3751:
3749:
3733:(53): 182–192,
3715:
3708:
3706:
3689:
3682:
3680:
3669:
3662:
3660:
3615:
3612:
3610:Primary sources
3607:
3602:
3595:Section 4.4 of
3594:
3590:
3542:
3541:
3537:
3530:
3526:
3518:
3509:
3480:
3479:
3472:
3460:
3456:
3448:
3441:
3435:Narkiewicz 2004
3432:
3428:
3420:
3416:
3410:Minkowski 1891b
3408:
3404:
3398:Minkowski 1891a
3396:
3392:
3384:
3380:
3372:
3368:
3360:
3356:
3349:
3345:
3328:
3321:
3313:
3309:
3301:
3297:
3289:
3285:
3277:
3273:
3262:
3258:Conrad, Keith.
3257:
3256:
3252:
3244:
3240:
3235:Washington 1997
3232:
3228:
3220:
3216:
3212:, Theorem 6.4.6
3208:
3204:
3198:Narkiewicz 2004
3196:
3192:
3184:
3180:
3175:Washington 1997
3172:
3168:
3160:
3156:
3134:
3118:
3117:
3110:
3103:
3099:
3095:
3054:Artin conductor
3025:
2992:
2973:
2965:
2960:
2959:
2927:
2915:
2914:
2912:
2877:
2869:
2868:
2864:
2848:tamely ramified
2836:
2834:
2822:
2820:
2797:
2765:
2755:
2729:
2728:
2696:
2686:
2660:
2659:
2646:
2638:
2603:
2589:
2565:) = (
2543:
2501:
2491:
2473:
2468:
2467:
2449:
2426:
2391:
2359:
2358:
2286:
2281:
2259:
2238:
2233:
2232:
2215:tower of fields
2212:
2202:
2155:
2148:
2134:
2125:
2118:
2109:
2090:
2081:
2059:
2050:
2031:
2022:
2003:
1997:
1993:
1984:
1975:
1966:
1949:
1926:
1887:
1875:
1853:
1786:
1785:
1775:
1765:
1748:
1729:
1728:
1718:
1708:
1683:
1673:
1663:
1662:
1650:
1564:
1559:
1558:
1544:
1535:if and only if
1515:
1503:Brill's theorem
1499:
1492:, respectively.
1482:
1471:
1388:
1378:
1365:
1329:
1328:
1320:
1314:
1309:associated to α
1300:
1287:
1280:
1273:
1266:
1241:
1185:
1184:
1127:
1089:
1060:
1036:
1031:
1026:
1025:
1017:
1004:
994:
977:
936:
935:
894:
885:
884:
852:
841:
828:
823:
822:
776:
775:
760:
753:
744:
716:
708:
699:
662:
633:
632:
619:
606:
604:
599:
594:
581:
568:
565:
564:
559:
553:
547:
546:
540:
535:
522:
509:
506:
505:
492:
479:
477:
472:
459:
446:
444:
431:
418:
410:
409:
393:
388:
387:
382:
373:
314:complex numbers
307:
301:
290:
277:
268:
257:
243:
227:principal ideal
92:is a numerical
59:
24:
17:
12:
11:
5:
4323:
4321:
4313:
4312:
4302:
4301:
4298:
4297:
4276:
4273:
4272:
4271:
4250:
4237:
4208:
4173:
4160:
4136:
4114:
4088:
4075:
4062:
4049:
4032:
4019:
4003:Taylor, Martin
3995:
3974:
3950:
3937:
3917:
3903:
3881:
3878:
3877:
3876:
3857:
3832:
3791:
3756:
3713:
3687:
3667:
3611:
3608:
3606:
3603:
3601:
3600:
3588:
3535:
3524:
3507:
3470:
3454:
3439:
3426:
3414:
3402:
3390:
3386:Kronecker 1882
3378:
3366:
3354:
3343:
3319:
3307:
3295:
3283:
3271:
3250:
3238:
3226:
3214:
3202:
3190:
3178:
3166:
3154:
3132:
3108:
3096:
3094:
3091:
3090:
3089:
3042:
3031:
3023:
3005:
2999:
2995:
2990:
2980:
2976:
2972:
2968:
2940:
2934:
2930:
2925:
2908:
2892:
2885:
2880:
2876:
2863:
2860:
2842:≈ 296.276, so
2796:
2793:
2789:
2788:
2777:
2772:
2768:
2762:
2758:
2754:
2751:
2748:
2745:
2742:
2739:
2736:
2718:
2717:
2703:
2699:
2693:
2689:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2642:
2633:
2599:
2585:
2577: × 2
2542:
2539:
2535:
2534:
2523:
2518:
2514:
2510:
2505:
2498:
2494:
2489:
2485:
2480:
2476:
2448:
2445:
2418:
2390:
2387:
2368:
2355:
2354:
2341:
2338:
2335:
2332:
2329:
2324:
2320:
2316:
2312:
2307:
2301:
2297:
2293:
2289:
2284:
2278:
2274:
2270:
2264:
2258:
2253:
2249:
2245:
2241:
2208:
2194:
2153:
2146:
2130:
2123:
2114:
2107:
2086:
2077:
2068:matrix whose (
2055:
2048:
2027:
2020:
1999:
1995:
1989:
1980:
1971:
1962:
1941:
1925:
1922:
1886:
1883:
1882:
1881:
1871:
1855:
1849:
1834:
1833:
1832:
1831:
1820:
1815:
1811:
1807:
1802:
1797:
1794:
1789:
1781:
1778:
1772:
1768:
1762:
1755:
1751:
1745:
1740:
1737:
1732:
1724:
1721:
1715:
1711:
1705:
1700:
1696:
1692:
1687:
1680:
1676:
1671:
1657:
1656:
1648:
1620:
1619:
1618:
1617:
1606:
1602:
1599:
1595:
1592:
1587:
1582: or
1579:
1576:
1571:
1567:
1553:
1552:
1546:
1540:
1525:
1518:complex places
1513:
1498:
1495:
1494:
1493:
1465:non-isomorphic
1457:
1456:is −503.
1414:
1413:
1410:
1409:
1408:
1395:
1391:
1385:
1381:
1377:
1372:
1368:
1364:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1337:
1323:
1322:
1316:
1315: = σ
1310:
1296:
1285:
1278:
1271:
1262:
1237:
1226:
1225:
1201:
1198:
1195:
1192:
1181:
1180:
1179:
1163:
1160:
1157:
1154:
1151:
1147:
1143:
1140:
1137:
1134:
1130:
1124:
1120:
1116:
1112:
1105:
1102:
1099:
1096:
1092:
1084:
1080:
1076:
1073:
1070:
1067:
1063:
1059:
1056:
1053:
1050:
1043:
1039:
1034:
1020:
1019:
1013:
1000:
990:
973:
962:
961:
954:
953:
952:
940:
934:
930:
927:
923:
920:
915:
912:
909:
906:
903:
895:
893:
890:
887:
886:
882:
879:
875:
872:
867:
864:
861:
853:
851:
848:
847:
844:
840:
835:
831:
817:
816:
804:
799:
794:
790:
786:
783:
759:
756:
749:
742:
737:integral basis
712:
704:
691:
660:
659:
648:
643:
638:
631:
626:
622:
618:
613:
609:
605:
603:
600:
598:
595:
593:
588:
584:
580:
575:
571:
567:
566:
563:
560:
558:
555:
552:
549:
548:
545:
542:
539:
536:
534:
529:
525:
521:
516:
512:
508:
507:
504:
499:
495:
491:
486:
482:
478:
476:
473:
471:
466:
462:
458:
453:
449:
445:
443:
438:
434:
430:
425:
421:
417:
416:
413:
408:
405:
400:
396:
378:
369:
303:
299:
286:
280:integral basis
273:
266:
253:
242:
239:
55:
34:obtained from
15:
13:
10:
9:
6:
4:
3:
2:
4322:
4311:
4308:
4307:
4305:
4289:
4288:
4283:
4279:
4278:
4274:
4269:
4265:
4261:
4257:
4253:
4247:
4243:
4238:
4235:
4231:
4227:
4223:
4219:
4215:
4211:
4205:
4201:
4197:
4192:
4187:
4183:
4179:
4174:
4171:
4167:
4163:
4161:0-12-163251-2
4157:
4153:
4149:
4145:
4141:
4137:
4133:
4129:
4125:
4121:
4117:
4111:
4107:
4102:
4097:
4093:
4089:
4086:
4082:
4078:
4072:
4068:
4063:
4060:
4056:
4052:
4050:3-540-63003-1
4046:
4042:
4038:
4033:
4030:
4026:
4022:
4016:
4012:
4008:
4004:
4000:
3996:
3993:
3989:
3985:
3981:
3977:
3971:
3967:
3963:
3959:
3955:
3951:
3948:
3944:
3940:
3934:
3930:
3926:
3922:
3918:
3914:
3910:
3906:
3900:
3896:
3895:Meldrum, John
3892:
3888:
3884:
3883:
3879:
3874:
3870:
3866:
3862:
3858:
3855:
3851:
3847:
3843:
3842:
3837:
3833:
3824:
3820:
3816:
3812:
3808:
3804:
3800:
3796:
3792:
3783:
3779:
3775:
3771:
3770:
3765:
3761:
3757:
3748:
3744:
3740:
3736:
3732:
3728:
3727:
3722:
3718:
3714:
3704:
3700:
3696:
3692:
3688:
3678:
3677:
3672:
3668:
3659:
3655:
3651:
3647:
3643:
3639:
3635:
3631:
3627:
3623:
3619:
3614:
3613:
3609:
3604:
3598:
3592:
3589:
3584:
3580:
3576:
3572:
3568:
3564:
3560:
3556:
3552:
3549:(in French).
3548:
3547:
3539:
3536:
3533:
3528:
3525:
3521:
3516:
3514:
3512:
3508:
3502:
3497:
3493:
3490:
3489:
3484:
3477:
3475:
3471:
3468:
3464:
3463:Neukirch 1999
3458:
3455:
3451:
3450:Neukirch 1999
3446:
3444:
3440:
3436:
3430:
3427:
3423:
3418:
3415:
3411:
3406:
3403:
3399:
3394:
3391:
3387:
3382:
3379:
3375:
3370:
3367:
3363:
3358:
3355:
3352:
3351:Bourbaki 1994
3347:
3344:
3340:
3339:Dedekind 1871
3336:
3332:
3326:
3324:
3320:
3317:
3316:Neukirch 1999
3311:
3308:
3305:
3304:Neukirch 1999
3299:
3296:
3293:
3292:Neukirch 1999
3287:
3284:
3281:
3280:Neukirch 1999
3275:
3272:
3268:
3261:
3254:
3251:
3248:
3247:Neukirch 1999
3242:
3239:
3236:
3233:Lemma 2.2 of
3230:
3227:
3223:
3218:
3215:
3211:
3206:
3203:
3199:
3194:
3191:
3187:
3186:Dedekind 1878
3182:
3179:
3176:
3170:
3167:
3164:
3158:
3155:
3151:
3147:
3143:
3139:
3135:
3129:
3125:
3121:
3120:Manin, Yu. I.
3115:
3113:
3109:
3106:
3101:
3098:
3092:
3087:
3084:, called the
3083:
3079:
3075:
3071:
3067:
3063:
3059:
3055:
3051:
3047:
3043:
3040:
3036:
3032:
3029:
3022:
2997:
2978:
2974:
2970:
2966:
2957:
2932:
2911:
2907:
2878:
2874:
2866:
2865:
2861:
2859:
2857:
2853:
2849:
2845:
2839:
2832:
2826:
2818:
2814:
2810:
2806:
2802:
2794:
2792:
2775:
2770:
2766:
2760:
2756:
2752:
2746:
2743:
2740:
2734:
2727:
2726:
2725:
2723:
2701:
2697:
2691:
2687:
2683:
2677:
2674:
2671:
2665:
2658:
2657:
2656:
2654:
2650:
2645:
2641:
2636:
2631:
2627:
2623:
2619:
2615:
2611:
2607:
2602:
2597:
2593:
2588:
2584:
2580:
2576:
2572:
2568:
2564:
2560:
2556:
2552:
2548:
2540:
2538:
2521:
2516:
2512:
2508:
2496:
2483:
2478:
2474:
2466:
2465:
2464:
2462:
2459:number field
2458:
2454:
2446:
2444:
2442:
2438:
2434:
2430:
2425:
2421:
2416:
2412:
2408:
2404:
2400:
2396:
2388:
2386:
2384:
2336:
2333:
2330:
2322:
2318:
2314:
2305:
2299:
2295:
2291:
2282:
2276:
2272:
2268:
2256:
2251:
2247:
2243:
2231:
2230:
2229:
2227:
2223:
2219:
2216:
2213: . In a
2211:
2206:
2201:
2197:
2192:
2188:
2184:
2180:
2176:
2172:
2168:
2164:
2160:
2156:
2150: ∈
2149:
2142:
2138:
2133:
2129:
2122:
2117:
2113:
2106:
2102:
2098:
2094:
2089:
2085:
2080:
2075:
2071:
2067:
2063:
2058:
2054:
2047:
2043:
2039:
2035:
2030:
2026:
2019:
2015:
2011:
2007:
2002:
1992:
1988:
1983:
1979:
1974:
1970:
1965:
1961:
1957:
1953:
1948:
1944:
1939:
1935:
1931:
1923:
1921:
1919:
1914:
1910:
1905:
1903:
1895:
1891:
1884:
1879:
1874:
1869:
1865:
1861:
1860:
1856:
1852:
1847:
1843:
1839:
1836:
1835:
1818:
1813:
1809:
1805:
1800:
1795:
1792:
1787:
1779:
1776:
1770:
1766:
1760:
1753:
1749:
1743:
1738:
1735:
1730:
1722:
1719:
1713:
1709:
1703:
1698:
1694:
1690:
1678:
1661:
1660:
1659:
1658:
1654:
1647:
1643:
1639:
1635:
1631:
1627:
1626:
1622:
1621:
1604:
1597:
1593:
1585:
1577:
1574:
1569:
1557:
1556:
1555:
1554:
1550:
1547:
1543:
1538:
1534:
1530:
1526:
1523:
1519:
1512:
1508:
1504:
1501:
1500:
1497:Basic results
1496:
1489:
1485:
1478:
1474:
1469:
1466:
1462:
1461:higher-degree
1458:
1455:
1451:
1447:
1443:
1439:
1436:
1432:
1428:
1424:
1420:
1416:
1415:
1411:
1393:
1383:
1379:
1375:
1370:
1366:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1335:
1327:
1326:
1325:
1324:
1319:
1313:
1308:
1304:
1301: =
1299:
1295:
1291:
1288: =
1284:
1277:
1270:
1265:
1261:
1257:
1253:
1249:
1245:
1240:
1236:
1232:
1228:
1227:
1223:
1219:
1215:
1196:
1190:
1182:
1158:
1155:
1152:
1145:
1138:
1132:
1128:
1122:
1114:
1110:
1100:
1094:
1090:
1082:
1078:
1071:
1065:
1057:
1054:
1048:
1041:
1037:
1024:
1023:
1022:
1021:
1016:
1012:
1008:
1003:
998:
993:
989:
985:
983:
976:
971:
967:
964:
963:
959:
955:
932:
925:
921:
913:
910:
907:
904:
901:
891:
888:
877:
873:
865:
862:
859:
849:
842:
838:
833:
821:
820:
819:
818:
797:
784:
781:
773:
769:
765:
762:
761:
757:
755:
752:
748:
741:
738:
734:
729:
727:
723:
720:
715:
711:
707:
703:
698:
694:
690:
686:
682:
678:
674:
670:
666:
646:
641:
636:
624:
620:
611:
607:
601:
596:
586:
582:
573:
569:
561:
556:
550:
543:
537:
527:
523:
514:
510:
497:
493:
484:
480:
474:
464:
460:
451:
447:
436:
432:
423:
419:
411:
403:
398:
386:
385:
384:
381:
377:
372:
367:
363:
359:
356:
353:
349:
345:
341:
337:
333:
329:
326: →
325:
322:
319:
315:
311:
306:
298:), and let {σ
297:
295:
289:
285:
281:
276:
272:
265:
261:
256:
252:
248:
240:
238:
236:
232:
228:
224:
220:
216:
213: =
212:
208:
204:
200:
196:
192:
188:
184:
180:
177:
173:
169:
165:
161:
156:
154:
150:
146:
142:
138:
134:
130:
126:
122:
117:
115:
111:
107:
103:
99:
95:
91:
87:
83:
75:
71:
67:
63:
58:
54: ⊗
53:
49:
45:
41:
37:
33:
28:
22:
4291:, retrieved
4286:
4241:
4181:
4151:
4095:
4066:
4036:
4006:
3957:
3954:Cohen, Henri
3924:
3921:Cohen, Henri
3890:
3864:
3845:
3839:
3826:, retrieved
3806:
3802:
3785:, retrieved
3773:
3767:
3750:, retrieved
3730:
3724:
3707:, retrieved
3702:
3698:
3681:, retrieved
3675:
3661:, retrieved
3628:(1): 87–89,
3625:
3621:
3591:
3550:
3544:
3538:
3527:
3491:
3486:
3457:
3429:
3417:
3405:
3393:
3381:
3369:
3362:Hermite 1857
3357:
3346:
3310:
3298:
3286:
3274:
3266:
3253:
3241:
3229:
3224:, p. 11
3217:
3205:
3193:
3181:
3169:
3157:
3123:
3100:
3081:
3077:
3069:
3065:
3062:Galois group
3049:
3045:
3034:
3027:
3020:
2909:
2905:
2855:
2851:
2847:
2843:
2837:
2830:
2824:
2816:
2812:
2808:
2804:
2798:
2790:
2719:
2652:
2648:
2643:
2639:
2634:
2629:
2625:
2621:
2617:
2613:
2609:
2605:
2600:
2595:
2591:
2586:
2582:
2578:
2574:
2570:
2566:
2562:
2558:
2554:
2550:
2546:
2544:
2536:
2460:
2456:
2455:of a degree
2452:
2450:
2437:class number
2432:
2428:
2423:
2419:
2414:
2413:ramifies in
2410:
2406:
2402:
2398:
2395:ramification
2392:
2389:Ramification
2356:
2225:
2221:
2217:
2209:
2204:
2199:
2195:
2190:
2186:
2182:
2178:
2166:
2162:
2158:
2151:
2144:
2140:
2136:
2131:
2127:
2120:
2115:
2111:
2104:
2100:
2096:
2092:
2087:
2083:
2078:
2076:)-entry is σ
2073:
2069:
2065:
2061:
2056:
2052:
2045:
2041:
2037:
2033:
2028:
2024:
2017:
2013:
2009:
2005:
2000:
1990:
1986:
1981:
1977:
1972:
1968:
1963:
1959:
1955:
1951:
1946:
1942:
1937:
1933:
1929:
1927:
1906:
1901:
1899:
1877:
1872:
1867:
1863:
1857:
1850:
1845:
1841:
1837:
1652:
1645:
1641:
1637:
1629:
1623:
1548:
1541:
1536:
1532:
1531:ramifies in
1528:
1521:
1510:
1502:
1487:
1483:
1476:
1472:
1468:cubic fields
1453:
1445:
1441:
1437:
1422:
1418:
1317:
1311:
1302:
1297:
1293:
1289:
1282:
1275:
1268:
1263:
1259:
1252:discriminant
1247:
1243:
1238:
1234:
1221:
1217:
1014:
1010:
1006:
1001:
996:
991:
987:
981:
974:
969:
767:
750:
746:
739:
730:
725:
721:
718:
713:
709:
705:
701:
696:
692:
688:
684:
680:
672:
668:
661:
379:
375:
370:
368:)-entry is σ
365:
361:
357:
351:
347:
335:
332:discriminant
331:
327:
323:
309:
304:
293:
287:
283:
274:
270:
263:
254:
250:
246:
244:
234:
230:
222:
218:
214:
210:
206:
202:
198:
194:
190:
182:
178:
171:
167:
163:
159:
157:
153:open problem
140:
132:
118:
86:discriminant
85:
79:
69:
65:
61:
56:
51:
47:
43:
39:
35:
31:
3848:: 209–212,
3532:Voight 2008
3494:: 415–423.
3188:, pp. 30–31
1632:denote the
1018:is given by
687:)-entry is
344:determinant
82:mathematics
4293:2008-08-20
4268:0966.11047
4234:1205.11125
4132:0956.11021
4059:0819.11044
3873:29.0172.03
3854:23.0214.01
3828:2009-08-20
3823:23.0212.01
3787:2009-08-20
3782:14.0038.02
3752:2009-08-20
3709:2009-08-20
3683:2009-08-05
3663:2009-08-22
3642:09.0059.02
3605:References
3597:Serre 1967
3583:0369.12007
3374:Brill 1877
3210:Cohen 1993
3163:Cohen 1993
3150:1079.11002
3074:characters
1490:− 35
1486:− 21
1475:− 21
1435:polynomial
986:, and let
980:primitive
735:in K with
677:trace form
241:Definition
135:, and the
4191:0802.0194
3984:0302-9743
3797:(1891a),
3776:: 1–122,
3747:120694650
3658:120947279
3575:122278145
3553:: 65–73.
3520:Koch 1997
3222:Koch 1997
3142:0938-0396
2994:Δ
2971:−
2929:Δ
2879:⊗
2771:σ
2761:ρ
2753:≥
2747:σ
2741:ρ
2735:α
2702:σ
2692:ρ
2684:≥
2678:σ
2672:ρ
2666:α
2655:). Then
2493:Δ
2311:Δ
2288:Δ
2240:Δ
2175:different
1985:basis of
1848:, then |Δ
1793:π
1761:≥
1736:π
1704:≥
1675:Δ
1575:≡
1566:Δ
1539:divides Δ
1433:α of the
1427:adjoining
1380:α
1376:−
1367:α
1355:≤
1343:≤
1336:∏
1220:dividing
1191:φ
1156:−
1133:φ
1111:∏
1095:φ
1066:φ
1055:−
1033:Δ
1005:) be the
905:≡
863:≡
830:Δ
608:σ
602:⋯
597:⋯
570:σ
562:⋮
557:⋱
551:⋮
544:⋮
538:⋱
511:σ
481:σ
475:⋯
448:σ
420:σ
395:Δ
318:injective
312:into the
176:extension
145:A theorem
94:invariant
4304:Category
4284:(1998),
4226:30036220
4150:(eds.),
4094:(1999).
4005:(1993),
3923:(1993),
3889:(1994).
3762:(1882),
3719:(1857),
3693:(1878),
3673:(1871),
3452:, §III.2
3019:, where
2720:and the
2637:→∞
2573:) is in
2157:for all
1998:, ..., σ
1994:. Let {σ
1930:absolute
1527:A prime
898:if
856:if
758:Examples
302:, ..., σ
121:analytic
114:ramified
74:ramified
72:is only
4260:1421575
4218:2467853
4170:0220701
4124:1697859
4085:2078267
4029:1215934
3992:2041075
3947:1228206
3913:1290116
3650:1509928
3555:Bibcode
3200:, p. 64
3060:of the
3056:of the
3052:is the
2956:measure
2835:√
2829:) with
2821:√
2811:,
2651:,
2628:,
2594:,
2185:. When
2173:of the
2169:is the
2126:, ...,
2110:, ...,
2051:, ...,
2023:, ...,
1885:History
1876:| <
1870:with |Δ
1844:is not
1545: .
1292:, ...,
1254:of the
745:, ...,
360:whose (
346:of the
342:of the
338:is the
330:). The
296:-module
269:, ...,
258:be its
225:is the
149:Hermite
127:of the
104:of the
4266:
4258:
4248:
4232:
4224:
4216:
4206:
4168:
4158:
4130:
4122:
4112:
4083:
4073:
4057:
4047:
4027:
4017:
3990:
3982:
3972:
3945:
3935:
3911:
3901:
3871:
3852:
3821:
3780:
3745:
3656:
3648:
3640:
3581:
3573:
3148:
3140:
3130:
2581:, let
2357:where
2119:) as {
2040:, let
1862:: Let
1655:, then
1634:degree
1628:: Let
1505:: The
1267:to be
1183:where
968:: let
766:: let
355:matrix
340:square
316:(i.e.
278:be an
262:. Let
174:of an
110:primes
88:of an
84:, the
60:
4222:S2CID
4186:arXiv
3743:S2CID
3654:S2CID
3571:S2CID
3263:(PDF)
3093:Notes
2757:215.3
2036:over
2016:. If
2008:into
1840:: If
978:be a
770:be a
733:order
667:from
665:trace
187:ideal
76:at 7.
4246:ISBN
4204:ISBN
4156:ISBN
4110:ISBN
4071:ISBN
4045:ISBN
4015:ISBN
3980:ISSN
3970:ISBN
3933:ISBN
3899:ISBN
3807:1891
3731:1857
3138:ISSN
3128:ISBN
2767:44.7
2698:22.3
2688:60.8
2549:and
2451:The
2383:norm
2171:norm
1644:and
1507:sign
1479:+ 28
1431:root
1417:Let
1349:<
245:Let
139:for
112:are
4264:Zbl
4230:Zbl
4196:doi
4128:Zbl
4055:Zbl
3962:doi
3869:JFM
3850:JFM
3846:112
3819:JFM
3811:doi
3778:JFM
3735:doi
3705:(1)
3638:JFM
3630:doi
3579:Zbl
3563:doi
3496:doi
3333:'s
3146:Zbl
3064:of
2913:is
2604:as
2409:of
2177:of
2064:by
1594:mod
1520:of
1481:or
1212:is
922:mod
874:mod
671:to
407:det
350:by
334:of
282:of
229:of
166:of
147:of
131:of
80:In
4306::
4262:,
4256:MR
4254:,
4228:,
4220:,
4214:MR
4212:,
4202:,
4194:,
4166:MR
4164:,
4146:;
4126:.
4120:MR
4118:.
4108:.
4098:.
4081:MR
4079:,
4053:,
4043:,
4025:MR
4023:,
4013:,
4001:;
3988:MR
3986:,
3978:,
3968:,
3943:MR
3941:,
3931:,
3909:MR
3907:.
3844:,
3817:,
3805:,
3801:,
3774:92
3772:,
3766:,
3741:,
3729:,
3723:,
3703:23
3701:,
3697:,
3652:,
3646:MR
3644:,
3636:,
3626:12
3624:,
3620:,
3577:.
3569:.
3561:.
3551:44
3510:^
3492:33
3485:.
3473:^
3442:^
3322:^
3265:.
3144:,
3136:,
3111:^
3030:).
2571:σn
2567:ρn
2561:,2
2475:rd
2385:.
2189:=
1429:a
1421:=
1242:=
999:(ζ
995:=
815:is
689:Tr
237:.
143:.
116:.
4198::
4188::
4134:.
3964::
3915:.
3813::
3737::
3632::
3585:.
3565::
3557::
3504:.
3498::
3424:.
3412:.
3400:.
3388:.
3376:.
3364:.
3341:)
3337:(
3088:.
3082:L
3080:/
3078:K
3070:L
3068:/
3066:K
3050:L
3048:/
3046:K
3041:.
3035:K
3028:K
3024:2
3021:r
3004:|
2998:K
2989:|
2979:2
2975:r
2967:2
2939:|
2933:K
2924:|
2910:K
2906:O
2891:R
2884:Q
2875:K
2856:α
2852:α
2844:α
2838:m
2831:m
2825:m
2823:-
2819:(
2817:Q
2813:σ
2809:ρ
2807:(
2805:α
2776:.
2750:)
2744:,
2738:(
2716:,
2681:)
2675:,
2669:(
2653:σ
2649:ρ
2647:(
2644:n
2640:α
2635:n
2630:σ
2626:ρ
2624:(
2622:α
2618:s
2614:r
2610:n
2606:K
2601:K
2596:σ
2592:ρ
2590:(
2587:n
2583:α
2579:Z
2575:Z
2569:,
2563:s
2559:r
2555:n
2551:σ
2547:ρ
2522:.
2517:n
2513:/
2509:1
2504:|
2497:K
2488:|
2484:=
2479:K
2461:K
2457:n
2433:Q
2429:Q
2424:L
2422:/
2420:K
2415:K
2411:L
2407:p
2403:L
2401:/
2399:K
2367:N
2340:]
2337:L
2334::
2331:K
2328:[
2323:F
2319:/
2315:L
2306:)
2300:L
2296:/
2292:K
2283:(
2277:F
2273:/
2269:L
2263:N
2257:=
2252:F
2248:/
2244:K
2226:F
2224:/
2222:L
2220:/
2218:K
2210:K
2205:Z
2200:Q
2198:/
2196:K
2191:Q
2187:L
2183:L
2181:/
2179:K
2167:L
2165:/
2163:K
2159:i
2154:K
2152:O
2147:i
2145:b
2141:L
2139:/
2137:K
2132:n
2128:b
2124:1
2121:b
2116:n
2112:b
2108:1
2105:b
2103:(
2101:d
2097:L
2095:/
2093:K
2088:j
2084:b
2082:(
2079:i
2074:j
2072:,
2070:i
2066:n
2062:n
2057:n
2053:b
2049:1
2046:b
2044:(
2042:d
2038:L
2034:K
2029:n
2025:b
2021:1
2018:b
2014:L
2010:C
2006:K
2001:n
1996:1
1991:K
1987:O
1982:L
1978:O
1973:L
1969:O
1964:L
1960:O
1956:L
1954:/
1952:K
1947:L
1945:/
1943:K
1940:Δ
1934:K
1902:K
1878:N
1873:K
1868:K
1864:N
1851:K
1846:Q
1842:K
1819:.
1814:2
1810:/
1806:n
1801:)
1796:4
1788:(
1780:!
1777:n
1771:n
1767:n
1754:2
1750:r
1744:)
1739:4
1731:(
1723:!
1720:n
1714:n
1710:n
1699:2
1695:/
1691:1
1686:|
1679:K
1670:|
1653:K
1649:2
1646:r
1642:Q
1640:/
1638:K
1630:n
1605:.
1601:)
1598:4
1591:(
1586:1
1578:0
1570:K
1551::
1542:K
1537:p
1533:K
1529:p
1524:.
1522:K
1514:2
1511:r
1488:x
1484:x
1477:x
1473:x
1454:K
1446:x
1442:x
1438:x
1423:Q
1419:K
1394:2
1390:)
1384:j
1371:i
1363:(
1358:n
1352:j
1346:i
1340:1
1318:i
1312:i
1303:α
1298:n
1294:b
1290:α
1286:3
1283:b
1279:2
1276:b
1272:1
1269:b
1264:K
1260:O
1248:K
1244:Z
1239:K
1235:O
1224:.
1222:n
1218:p
1200:)
1197:n
1194:(
1162:)
1159:1
1153:p
1150:(
1146:/
1142:)
1139:n
1136:(
1129:p
1123:n
1119:|
1115:p
1104:)
1101:n
1098:(
1091:n
1083:2
1079:/
1075:)
1072:n
1069:(
1062:)
1058:1
1052:(
1049:=
1042:n
1038:K
1015:n
1011:K
1007:n
1002:n
997:Q
992:n
988:K
982:n
975:n
970:n
960:.
933:.
929:)
926:4
919:(
914:3
911:,
908:2
902:d
892:d
889:4
881:)
878:4
871:(
866:1
860:d
850:d
843:{
839:=
834:K
803:)
798:d
793:(
789:Q
785:=
782:K
768:d
751:n
747:b
743:1
740:b
726:K
722:B
719:B
714:j
710:b
706:i
702:b
700:(
697:Q
695:/
693:K
685:j
683:,
681:i
673:Q
669:K
647:.
642:2
637:)
630:)
625:n
621:b
617:(
612:n
592:)
587:1
583:b
579:(
574:n
533:)
528:1
524:b
520:(
515:2
503:)
498:n
494:b
490:(
485:1
470:)
465:2
461:b
457:(
452:1
442:)
437:1
433:b
429:(
424:1
412:(
404:=
399:K
380:j
376:b
374:(
371:i
366:j
364:,
362:i
358:B
352:n
348:n
336:K
328:C
324:K
310:K
305:n
300:1
294:Z
288:K
284:O
275:n
271:b
267:1
264:b
255:K
251:O
247:K
235:K
231:Z
223:Q
221:/
219:K
215:Q
211:L
207:Q
203:L
199:L
197:/
195:K
191:L
183:L
181:/
179:K
168:K
160:K
141:K
133:K
70:K
66:K
62:R
57:Q
52:K
48:x
44:x
40:x
36:Q
32:K
23:.
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