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对于任意 理想 I戴德金环 中,存在一个 理想 I_i 使得

 II_i=z,
(1)

其中 z 是一个 主理想,(即,秩为 1 的 理想)。此外,对于具有有限理想类群的 戴德金环,存在一个有限的理想列表 I_i 使得对于某些 I,此方程可以满足。此列表的大小被称为类数。

类数通常在 数域 的阶的上下文中研究。如果此阶是极大的,则它是 数域 的整数环,在这种情况下,类数等于 数域类群 的阶;否则,它等于所讨论的非极大阶的 皮卡德群 的阶。

数域整数环 的类数为 1 时,对应于给定 理想 的环具有唯一分解,并且在某种意义上,类数是该环中唯一分解失败程度的度量。

给出 的精确类数的有限级数被称为 类数公式。对于分圆整数的完整环以及分圆整数的任何子环,都已知 类数公式。找到类数是一个计算上困难的问题。

Wolfram 语言 函数NumberFieldClassNumber[Sqrt[d]] 给出 h(d) 的类数 d,其中 d基本判别式

判别式为 d<0 的二次域的阶的类数 h(d) 等于判别式为 d 的约化二元二次型的数量。例如,数域 Q(-sqrt(23)) 的整数环 Z((1+sqrt(-23))/2) 的类数 h(-23) 等于 3,因为存在三个判别式为 -23 的约化二元二次型,即 <1,1,6><2,1,3><2,-1,3>。计算判别式为 d<0 的二次数域 Q(sqrt(d)) 的唯一阶的类数 h(d) 的一种低效方法是计算判别式为 d 的约化二元二次型的数量。

一些相当复杂的数学表明,判别式为 d 的类数可以由 类数公式 给出

 h(d)={-1/(2lneta(d))sum_(r=1)^(d-1)(d/r)lnsin((pir)/d)   for d>0; -(w(d))/(2|d|)sum_(r=1)^(|d|-1)(d/r)r   for d<0,
(2)

其中 (d/r)克罗内克符号eta(d)基本单位w(d) 是使 二元二次型 不变的替换次数

 w(d)={6   for d=-3; 4   for d=-4; 2   otherwise,
(3)

并且总和是对 克罗内克符号 被定义的所有项求和 (Cohn 1980)。对于 d>0,类数也可以写成

 eta^(2h(d))=product_(r=1)^(d-1)sin^(-(d/r))((pir)/d)
(4)

对于 d>0,其中 乘积 是对 克罗内克符号 被定义的项取乘积。

类数 h(d)狄利克雷 L 级数 相关,关系如下:

 h(d)=(L_d(1))/(kappa(d)),
(5)

其中 kappa(d)狄利克雷结构常数

Oesterlé (1985) 表明,类数 h(-d) 满足 不等式

 h(-d)>1/(7000)product_(p|d)^*(1-(|_2sqrt(p)_|)/(p+1))lnd,
(6)

对于 -d<0,其中 |_x_|向下取整函数,乘积是对整除 d素数 求积,* 表示从乘积中省略了 d最大素因子。还已知,如果 d 与 5077 互质,则 (6) 中的分母 7000 可以替换为 55。

高斯类数问题 要求确定基本 二元二次型判别式 -d 的完整列表,使得对于给定的 n,类数由 h(-d)=n 给出。对于 n<=7奇数 n<=23,此问题已得到解决。高斯推测 虚二次域 的类数 h(-d) 随着 二元二次型判别式 -d 中的 d 趋于无穷大而趋于无穷大,这一论断现在被称为 高斯类数猜想

对应于虚二次域的负判别式 d 为 3, 4, 7, 8, 11, 15, 19, 20, 23, 24, 31, 35, 39, 40, 43, ... (OEIS A003657),它们对应的类数 h(-d)=1 为 1, 1, 1, 1, 1, 2, 1, 2, 3, 2, 3, 2, 4, 2, 1, ... (OEIS A006641)。

已知具有类数 1-5 和 奇数 7-23 的完整负判别式集。Buell (1977) 给出了 基本判别式 的最小和最大类数,其中 d<4000000,分为 偶数 判别式、判别式 1 (mod 8) 和判别式 5 (mod 8)。Arno et al. (1993) 给出了 奇数 k=5、7、9、...、23 的 h(-d)=k 的基本 d 值的完整列表。Wagner (1996) 给出了 k=5、6 和 7 的完整值列表。下表给出了对应于具有小类数 h(-d)虚二次域 Q(sqrt(-d(n))) 基本判别式 列表。在该表中,N 是具有给定类数 h(-d)-d基本 值的数量,其中“基本”意味着 -d 不可被任何 平方数 s^2 整除,使得 h(-d/s^2)<h(-d)。例如,尽管 h(-63)=2,但 -63 不是基本判别式,因为 63=3^2·7h(-63/3^2)=h(-7)=1<h(-63)偶数8<=h(-d)<=24 已由 Weisstein 计算得出。

下表列出了具有类数 h<=25基本判别式 d (Cohen 1993, pp. 229 和 514-515; Cox 1997, p. 271)。对于类数 18、20、22 和 24,搜索分别在 50000、70000、90000 和 90000 处终止。据我所知,目前尚不知道这些情况的解析上限。

h(-d)N斯隆d
19A0146023, 4, 7, 8, 11, 19, 43, 67, 163
218A01460315, 20, 24, 35, 40, 51, 52, 88, 91, 115, 123, 148, 187, 232, 235, 267, 403, 427
316A00620323, 31, 59, 83, 107, 139, 211, 283, 307, 331, 379, 499, 547, 643, 883, 907
454A01365839, 55, 56, 68, 84, 120, 132, 136, 155, 168, 184, 195, 203, 219, 228, 259, 280, 291, 292, 312, 323, 328, 340, 355, 372, 388, 408, 435, 483, 520, 532, 555, 568, 595, 627, 667, 708, 715, 723, 760, 763, 772, 795, 955, 1003, 1012, 1027, 1227, 1243, 1387, 1411, 1435, 1507, 1555
525A04600247, 79, 103, 127, 131, 179, 227, 347, 443, 523, 571, 619, 683, 691, 739, 787, 947, 1051, 1123, 1723, 1747, 1867, 2203, 2347, 2683
651A04600387, 104, 116, 152, 212, 244, 247, 339, 411, 424, 436, 451, 472, 515, 628, 707, 771, 808, 835, 843, 856, 1048, 1059, 1099, 1108, 1147, 1192, 1203, 1219, 1267, 1315, 1347, 1363, 1432, 1563, 1588, 1603, 1843, 1915, 1963, 2227, 2283, 2443, 2515, 2563, 2787, 2923, 3235, 3427, 3523, 3763
731A04600471, 151, 223, 251, 463, 467, 487, 587, 811, 827, 859, 1163, 1171, 1483, 1523, 1627, 1787, 1987, 2011, 2083, 2179, 2251, 2467, 2707, 3019, 3067, 3187, 3907, 4603, 5107, 5923
8131A04600595, 111, 164, 183, 248, 260, 264, 276, 295, 299, 308, 371, 376, 395, 420, 452, 456, 548, 552, 564, 579, 580, 583, 616, 632, 651, 660, 712, 820, 840, 852, 868, 904, 915, 939, 952, 979, 987, 995, 1032, 1043, 1060, 1092, 1128, 1131, 1155, 1195, 1204, 1240, 1252, 1288, 1299, 1320, 1339, 1348, 1380, 1428, 1443, 1528, 1540, 1635, 1651, 1659, 1672, 1731, 1752, 1768, 1771, 1780, 1795, 1803, 1828, 1848, 1864, 1912, 1939, 1947, 1992, 1995, 2020, 2035, 2059, 2067, 2139, 2163, 2212, 2248, 2307, 2308, 2323, 2392, 2395, 2419, 2451, 2587, 2611, 2632, 2667, 2715, 2755, 2788, 2827, 2947, 2968, 2995, 3003, 3172, 3243, 3315, 3355, 3403, 3448, 3507, 3595, 3787, 3883, 3963, 4123, 4195, 4267, 4323, 4387, 4747, 4843, 4867, 5083, 5467, 5587, 5707, 5947, 6307
934A046006199, 367, 419, 491, 563, 823, 1087, 1187, 1291, 1423, 1579, 2003, 2803, 3163, 3259, 3307, 3547, 3643, 4027, 4243, 4363, 4483, 4723, 4987, 5443, 6043, 6427, 6763, 6883, 7723, 8563, 8803, 9067, 10627
1087A046007119, 143, 159, 296, 303, 319, 344, 415, 488, 611, 635, 664, 699, 724, 779, 788, 803, 851, 872, 916, 923, 1115, 1268, 1384, 1492, 1576, 1643, 1684, 1688, 1707, 1779, 1819, 1835, 1891, 1923, 2152, 2164, 2363, 2452, 2643, 2776, 2836, 2899, 3028, 3091, 3139, 3147, 3291, 3412, 3508, 3635, 3667, 3683, 3811, 3859, 3928, 4083, 4227, 4372, 4435, 4579, 4627, 4852, 4915, 5131, 5163, 5272, 5515, 5611, 5667, 5803, 6115, 6259, 6403, 6667, 7123, 7363, 7387, 7435, 7483, 7627, 8227, 8947, 9307, 10147, 10483, 13843
1141A046008167, 271, 659, 967, 1283, 1303, 1307, 1459, 1531, 1699, 2027, 2267, 2539, 2731, 2851, 2971, 3203, 3347, 3499, 3739, 3931, 4051, 5179, 5683, 6163, 6547, 7027, 7507, 7603, 7867, 8443, 9283, 9403, 9643, 9787, 10987, 13003, 13267, 14107, 14683, 15667
12206A046009231, 255, 327, 356, 440, 516, 543, 655, 680, 687, 696, 728, 731, 744, 755, 804, 888, 932, 948, 964, 984, 996, 1011, 1067, 1096, 1144, 1208, 1235, 1236, 1255, 1272, 1336, 1355, 1371, 1419, 1464, 1480, 1491, 1515, 1547, 1572, 1668, 1720, 1732, 1763, 1807, 1812, 1892, 1955, 1972, 2068, 2091, 2104, 2132, 2148, 2155, 2235, 2260, 2355, 2387, 2388, 2424, 2440, 2468, 2472, 2488, 2491, 2555, 2595, 2627, 2635, 2676, 2680, 2692, 2723, 2728, 2740, 2795, 2867, 2872, 2920, 2955, 3012, 3027, 3043, 3048, 3115, 3208, 3252, 3256, 3268, 3304, 3387, 3451, 3459, 3592, 3619, 3652, 3723, 3747, 3768, 3796, 3835, 3880, 3892, 3955, 3972, 4035, 4120, 4132, 4147, 4152, 4155, 4168, 4291, 4360, 4411, 4467, 4531, 4552, 4555, 4587, 4648, 4699, 4708, 4755, 4771, 4792, 4795, 4827, 4888, 4907, 4947, 4963, 5032, 5035, 5128, 5140, 5155, 5188, 5259, 5299, 5307, 5371, 5395, 5523, 5595, 5755, 5763, 5811, 5835, 6187, 6232, 6235, 6267, 6283, 6472, 6483, 6603, 6643, 6715, 6787, 6843, 6931, 6955, 6963, 6987, 7107, 7291, 7492, 7555, 7683, 7891, 7912, 8068, 8131, 8155, 8248, 8323, 8347, 8395, 8787, 8827, 9003, 9139, 9355, 9523, 9667, 9843, 10003, 10603, 10707, 10747, 10795, 10915, 11155, 11347, 11707, 11803, 12307, 12643, 14443, 15163, 15283, 16003, 17803
1337A046010191, 263, 607, 631, 727, 1019, 1451, 1499, 1667, 1907, 2131, 2143, 2371, 2659, 2963, 3083, 3691, 4003, 4507, 4643, 5347, 5419, 5779, 6619, 7243, 7963, 9547, 9739, 11467, 11587, 11827, 11923, 12043, 14347, 15787, 16963, 20563
1495A046011215, 287, 391, 404, 447, 511, 535, 536, 596, 692, 703, 807, 899, 1112, 1211, 1396, 1403, 1527, 1816, 1851, 1883, 2008, 2123, 2147, 2171, 2335, 2427, 2507, 2536, 2571, 2612, 2779, 2931, 2932, 3112, 3227, 3352, 3579, 3707, 3715, 3867, 3988, 4187, 4315, 4443, 4468, 4659, 4803, 4948, 5027, 5091, 5251, 5267, 5608, 5723, 5812, 5971, 6388, 6499, 6523, 6568, 6979, 7067, 7099, 7147, 7915, 8035, 8187, 8611, 8899, 9115, 9172, 9235, 9427, 10123, 10315, 10363, 10411, 11227, 12147, 12667, 12787, 13027, 13435, 13483, 13603, 14203, 16867, 18187, 18547, 18643, 20227, 21547, 23083, 30067
1568A046012239, 439, 751, 971, 1259, 1327, 1427, 1567, 1619, 2243, 2647, 2699, 2843, 3331, 3571, 3803, 4099, 4219, 5003, 5227, 5323, 5563, 5827, 5987, 6067, 6091, 6211, 6571, 7219, 7459, 7547, 8467, 8707, 8779, 9043, 9907, 10243, 10267, 10459, 10651, 10723, 11083, 11971, 12163, 12763, 13147, 13963, 14323, 14827, 14851, 15187, 15643, 15907, 16603, 16843, 17467, 17923, 18043, 18523, 19387, 19867, 20707, 22003, 26203, 27883, 29947, 32323, 34483
16322A046013399, 407, 471, 559, 584, 644, 663, 740, 799, 884, 895, 903, 943, 1015, 1016, 1023, 1028, 1047, 1139, 1140, 1159, 1220, 1379, 1412, 1416, 1508, 1560, 1595, 1608, 1624, 1636, 1640, 1716, 1860, 1876, 1924, 1983, 2004, 2019, 2040, 2056, 2072, 2095, 2195, 2211, 2244, 2280, 2292, 2296, 2328, 2356, 2379, 2436, 2568, 2580, 2584, 2739, 2760, 2811, 2868, 2884, 2980, 3063, 3108, 3140, 3144, 3160, 3171, 3192, 3220, 3336, 3363, 3379, 3432, 3435, 3443, 3460, 3480, 3531, 3556, 3588, 3603, 3640, 3732, 3752, 3784, 3795, 3819, 3828, 3832, 3939, 3976, 4008, 4020, 4043, 4171, 4179, 4180, 4216, 4228, 4251, 4260, 4324, 4379, 4420, 4427, 4440, 4452, 4488, 4515, 4516, 4596, 4612, 4683, 4687, 4712, 4740, 4804, 4899, 4939, 4971, 4984, 5115, 5160, 5187, 5195, 5208, 5363, 5380, 5403, 5412, 5428, 5460, 5572, 5668, 5752, 5848, 5860, 5883, 5896, 5907, 5908, 5992, 5995, 6040, 6052, 6099, 6123, 6148, 6195, 6312, 6315, 6328, 6355, 6395, 6420, 6532, 6580, 6595, 6612, 6628, 6708, 6747, 6771, 6792, 6820, 6868, 6923, 6952, 7003, 7035, 7051, 7195, 7288, 7315, 7347, 7368, 7395, 7480, 7491, 7540, 7579, 7588, 7672, 7707, 7747, 7755, 7780, 7795, 7819, 7828, 7843, 7923, 7995, 8008, 8043, 8052, 8083, 8283, 8299, 8308, 8452, 8515, 8547, 8548, 8635, 8643, 8680, 8683, 8715, 8835, 8859, 8932, 8968, 9208, 9219, 9412, 9483, 9507, 9508, 9595, 9640, 9763, 9835, 9867, 9955, 10132, 10168, 10195, 10203, 10227, 10312, 10387, 10420, 10563, 10587, 10635, 10803, 10843, 10948, 10963, 11067, 11092, 11107, 11179, 11203, 11512, 11523, 11563, 11572, 11635, 11715, 11848, 11995, 12027, 12259, 12387, 12523, 12595, 12747, 12772, 12835, 12859, 12868, 13123, 13192, 13195, 13288, 13323, 13363, 13507, 13795, 13819, 13827, 14008, 14155, 14371, 14403, 14547, 14707, 14763, 14995, 15067, 15387, 15403, 15547, 15715, 16027, 16195, 16347, 16531, 16555, 16723, 17227, 17323, 17347, 17427, 17515, 18403, 18715, 18883, 18907, 19147, 19195, 19947, 19987, 20155, 20395, 21403, 21715, 21835, 22243, 22843, 23395, 23587, 24403, 25027, 25267, 27307, 27787, 28963, 31243
1745A046014383, 991, 1091, 1571, 1663, 1783, 2531, 3323, 3947, 4339, 4447, 4547, 4651, 5483, 6203, 6379, 6451, 6827, 6907, 7883, 8539, 8731, 9883, 11251, 11443, 12907, 13627, 14083, 14779, 14947, 16699, 17827, 18307, 19963, 21067, 23563, 24907, 25243, 26083, 26107, 27763, 31627, 33427, 36523, 37123
18150A046015335, 519, 527, 679, 1135, 1172, 1207, 1383, 1448, 1687, 1691, 1927, 2047, 2051, 2167, 2228, 2291, 2315, 2344, 2644, 2747, 2859, 3035, 3107, 3543, 3544, 3651, 3688, 4072, 4299, 4307, 4568, 4819, 4883, 5224, 5315, 5464, 5492, 5539, 5899, 6196, 6227, 6331, 6387, 6484, 6739, 6835, 7323, 7339, 7528, 7571, 7715, 7732, 7771, 7827, 8152, 8203, 8212, 8331, 8403, 8488, 8507, 8587, 8884, 9123, 9211, 9563, 9627, 9683, 9748, 9832, 10228, 10264, 10347, 10523, 11188, 11419, 11608, 11643, 11683, 11851, 11992, 12067, 12148, 12187, 12235, 12283, 12651, 12723, 12811, 12952, 13227, 13315, 13387, 13747, 13947, 13987, 14163, 14227, 14515, 14667, 14932, 15115, 15243, 16123, 16171, 16387, 16627, 17035, 17131, 17403, 17635, 18283, 18712, 19027, 19123, 19651, 20035, 20827, 21043, 21652, 21667, 21907, 22267, 22443, 22507, 22947, 23347, 23467, 23683, 23923, 24067, 24523, 24667, 24787, 25435, 26587, 26707, 28147, 29467, 32827, 33763, 34027, 34507, 36667, 39307, 40987, 41827, 43387, 48427
1947A046016311, 359, 919, 1063, 1543, 1831, 2099, 2339, 2459, 3343, 3463, 3467, 3607, 4019, 4139, 4327, 5059, 5147, 5527, 5659, 6803, 8419, 8923, 8971, 9619, 10891, 11299, 15091, 15331, 16363, 16747, 17011, 17299, 17539, 17683, 19507, 21187, 21211, 21283, 23203, 24763, 26227, 27043, 29803, 31123, 37507, 38707
20350A046017455, 615, 776, 824, 836, 920, 1064, 1124, 1160, 1263, 1284, 1460, 1495, 1524, 1544, 1592, 1604, 1652, 1695, 1739, 1748, 1796, 1880, 1887, 1896, 1928, 1940, 1956, 2136, 2247, 2360, 2404, 2407, 2483, 2487, 2532, 2552, 2596, 2603, 2712, 2724, 2743, 2948, 2983, 2987, 3007, 3016, 3076, 3099, 3103, 3124, 3131, 3155, 3219, 3288, 3320, 3367, 3395, 3496, 3512, 3515, 3567, 3655, 3668, 3684, 3748, 3755, 3908, 3979, 4011, 4015, 4024, 4036, 4148, 4264, 4355, 4371, 4395, 4403, 4408, 4539, 4548, 4660, 4728, 4731, 4756, 4763, 4855, 4891, 5019, 5028, 5044, 5080, 5092, 5268, 5331, 5332, 5352, 5368, 5512, 5560, 5592, 5731, 5944, 5955, 5956, 5988, 6051, 6088, 6136, 6139, 6168, 6280, 6339, 6467, 6504, 6648, 6712, 6755, 6808, 6856, 7012, 7032, 7044, 7060, 7096, 7131, 7144, 7163, 7171, 7192, 7240, 7428, 7432, 7467, 7572, 7611, 7624, 7635, 7651, 7667, 7720, 7851, 7876, 7924, 7939, 8067, 8251, 8292, 8296, 8355, 8404, 8472, 8491, 8632, 8692, 8755, 8808, 8920, 8995, 9051, 9124, 9147, 9160, 9195, 9331, 9339, 9363, 9443, 9571, 9592, 9688, 9691, 9732, 9755, 9795, 9892, 9976, 9979, 10027, 10083, 10155, 10171, 10291, 10299, 10308, 10507, 10515, 10552, 10564, 10819, 10888, 11272, 11320, 11355, 11379, 11395, 11427, 11428, 11539, 11659, 11755, 11860, 11883, 11947, 11955, 12019, 12139, 12280, 12315, 12328, 12331, 12355, 12363, 12467, 12468, 12472, 12499, 12532, 12587, 12603, 12712, 12883, 12931, 12955, 12963, 13155, 13243, 13528, 13555, 13588, 13651, 13803, 13960, 14307, 14331, 14467, 14491, 14659, 14755, 14788, 15235, 15268, 15355, 15603, 15688, 15691, 15763, 15883, 15892, 15955, 16147, 16228, 16395, 16408, 16435, 16483, 16507, 16612, 16648, 16683, 16707, 16915, 16923, 17067, 17187, 17368, 17563, 17643, 17763, 17907, 18067, 18163, 18195, 18232, 18355, 18363, 19083, 19443, 19492, 19555, 19923, 20083, 20203, 20587, 20683, 20755, 20883, 21091, 21235, 21268, 21307, 21387, 21508, 21595, 21723, 21763, 21883, 22387, 22467, 22555, 22603, 22723, 23443, 23947, 24283, 24355, 24747, 24963, 25123, 25363, 26635, 26755, 26827, 26923, 27003, 27955, 27987, 28483, 28555, 29107, 29203, 30283, 30787, 31003, 31483, 31747, 31987, 32923, 33163, 34435, 35683, 35995, 36283, 37627, 37843, 37867, 38347, 39187, 39403, 40243, 40363, 40555, 40723, 43747, 47083, 48283, 51643, 54763, 58507
2185A046018431, 503, 743, 863, 1931, 2503, 2579, 2767, 2819, 3011, 3371, 4283, 4523, 4691, 5011, 5647, 5851, 5867, 6323, 6691, 7907, 8059, 8123, 8171, 8243, 8387, 8627, 8747, 9091, 9187, 9811, 9859, 10067, 10771, 11731, 12107, 12547, 13171, 13291, 13339, 13723, 14419, 14563, 15427, 16339, 16987, 17107, 17707, 17971, 18427, 18979, 19483, 19531, 19819, 20947, 21379, 22027, 22483, 22963, 23227, 23827, 25603, 26683, 27427, 28387, 28723, 28867, 31963, 32803, 34147, 34963, 35323, 36067, 36187, 39043, 40483, 44683, 46027, 49603, 51283, 52627, 55603, 58963, 59467, 61483
22139A171724591, 623, 767, 871, 879, 1076, 1111, 1167, 1304, 1556, 1591, 1639, 1903, 2215, 2216, 2263, 2435, 2623, 2648, 2815, 2863, 2935, 3032, 3151, 3316, 3563, 3587, 3827, 4084, 4115, 4163, 4328, 4456, 4504, 4667, 4811, 5383, 5416, 5603, 5716, 5739, 5972, 6019, 6127, 6243, 6616, 6772, 6819, 7179, 7235, 7403, 7763, 7768, 7899, 8023, 8143, 8371, 8659, 8728, 8851, 8907, 8915, 9267, 9304, 9496, 10435, 10579, 10708, 10851, 11035, 11283, 11363, 11668, 12091, 12115, 12403, 12867, 13672, 14019, 14059, 14179, 14548, 14587, 14635, 15208, 15563, 15832, 16243, 16251, 16283, 16291, 16459, 17147, 17587, 17779, 17947, 18115, 18267, 18835, 18987, 19243, 19315, 19672, 20308, 20392, 22579, 22587, 22987, 24243, 24427, 25387, 25507, 25843, 25963, 26323, 26548, 27619, 28267, 29227, 29635, 29827, 30235, 30867, 31315, 33643, 33667, 34003, 34387, 35347, 41083, 43723, 44923, 46363, 47587, 47923, 49723, 53827, 77683, 85507
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类数为 1、2、3、... 的负判别式的数量分别为 9、18、16、54、25、51、31、... (OEIS A046125)。类数为 1、2、3、... 的最大负判别式分别为 163、427、907、1555、2683、... (OEIS A038552)。

对应于实二次域的判别式 d 为 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40, 41, 44, 53, ... (OEIS A003658),对应的类数 h(d)=1 为 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, ... (OEIS A003652)。

下表给出了具有小类数 h(d) 的前几个 基本判别式 d 列表,这些判别式对应于 实二次域,并通过包含可被 4 整除的项来扩充 Cohn (1980, pp. 271-274) 的表格 (Cohen 1993, pp. 516-519; Cohen 2000, pp. 534-537)。事实上,除了可能的因子 4 之外,所有二次数域的判别式都是无平方因子数。

h(d)OEISd
1A0036565, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 41, 44, 53, 56, 57, 61, ...
2A09461940, 60, 65, 85, 104, 105, 120, 136, 140, 156, 165, 168, 185, 204, ...
3A094612229, 257, 316, 321, 469, 473, 568, 733, 761, 892, 993, 1016, 1101, ...
4A094613145, 328, 445, 505, 520, 680, 689, 777, 780, 793, 840, 876, 897, 901, ...
5A094614401, 817, 1093, 1393, 1429, 1641, 1756, 1897, 1996, 2081, 2153, 2908, ...

使得判别式为 d 的实二次域具有类 h(d)=n 的最小 d,对于 n=1、2、... 分别为 5、40、229、145、401、697、577、904、1129、... (OEIS A081364)。


另请参阅

二元二次型判别式, 类域论, 类群, 类数公式, 狄利克雷 L 级数, 狄利克雷结构常数, 高斯类数猜想, 高斯类数问题, 黑格纳数, 理想, j 函数,

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参考文献

Arno, S. "The Imaginary Quadratic Fields of Class Number 4." Acta Arith. 40, 321-334, 1992.Arno, S.; Robinson, M. L.; and Wheeler, F. S. "Imaginary Quadratic Fields with Small Odd Class Number." http://www.math.uiuc.edu/Algebraic-Number-Theory/0009/.Buell, D. A. "Small Class Numbers and Extreme Values of L-Functions of Quadratic Fields." Math. Comput. 139, 786-796, 1977.Cohen, H. "Table of Class Numbers of Complex Quadratic Fields" and "Table of Class Numbers and Units of Real Quadratic Fields." §B.1 和 B.2 in A Course in Computational Algebraic Number Theory. New York: Springer-Verlag, pp. 513-519, 1993.Cohen, H. "Hilbert Class Fields of Quadratic Fields." §12.1 in Advanced Topics in Computational Number Theory. New York: Springer-Verlag, pp. 533-542, 2000.Cohn, H. Advanced Number Theory. New York: Dover, pp. 163 和 234, 1980.Cox, D. A. Primes of the Form x2+ny2: Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1997.Davenport, H. "Dirichlet's Class Number Formula." Ch. 6 in Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, pp. 43-53, 1980.Finch, S. R. "Class Number Theory." http://algo.inria.fr/csolve/clss.pdf.Himmetoglu, S. Berechnung von Klassenzahlen Imaginaer-Quadratischer Zahlkörper. Diplomarbeit. Heidelberg, Germany: University of Heidelberg Faculty for Mathematics, March 1986.Iyanaga, S. 和 Kawada, Y. (Eds.). "Class Numbers of Algebraic Number Fields." Appendix B, Table 4 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1494-1496, 1980.Montgomery, H. 和 Weinberger, P. "Notes on Small Class Numbers." Acta. Arith. 24, 529-542, 1974.Müller, H. "A Calculation of Class-Numbers of Imaginary Quadratic Numberfields." Tamkang J. Math. 9, 121-128, 1978.Oesterlé, J. "Nombres de classes des corps quadratiques imaginaires." Astérique 121-122, 309-323, 1985.Sloane, N. J. A. Sequences A003652/M0051, A003656/M3777, A003657/M2332, A003658/M3776, A006203/M5131, A006641/M0112, A013658, A014602, A014603, A035120, A038552, A046002, A046003, A046125, A048925, A056987, A081364, A094612, A094613, A094614, 和 A094619 in "The On-Line Encyclopedia of Integer Sequences."Stark, H. M. "A Complete Determination of the Complex Quadratic Fields of Class Number One." Michigan Math. J. 14, 1-27, 1967.Stark, H. M. "On Complex Quadratic Fields with Class Number Two." Math. Comput. 29, 289-302, 1975.Wagner, C. "Class Number 5, 6, and 7." Math. Comput. 65, 785-800, 1996.

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类数

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韦斯坦因,埃里克·W. "类数。" 来自 MathWorld--Wolfram Web 资源。 https://mathworld.net.cn/ClassNumber.html

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