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Class groups of number fields: numerical heuristics


Authors: H. Cohen and J. Martinet
Journal: Math. Comp. 48 (1987), 123-137
MSC: Primary 11R29; Secondary 11Y40
DOI: https://doi.org/10.1090/S0025-5718-1987-0866103-4
MathSciNet review: 866103
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Abstract: Extending previous work of H. W. Lenstra, Jr. and the first author, we give quantitative conjectures for the statistical behavior of class groups and class numbers for every type of field of degree less than or equal to four (given the signature and the Galois group of the Galois closure). The theoretical justifications for these conjectures will appear elsewhere, but the agreement with the existing tables is quite good.


References [Enhancements On Off] (What's this?)

  • [1] I. O. Angell, "A table of complex cubic fields," Bull. London Math. Soc., v. 5, 1973, pp. 37-38. MR 0318099 (47:6648)
  • [2] D. A. Buell, "The expectation of success using a Monte-Carlo factoring method--Some statistics on quadratic class numbers," Math. Comp., v. 43, 1984, pp. 313-327. MR 744940 (85k:11068)
  • [3] H. Cohen & H. W. Lenstra, Jr., "Heuristics on class groups of number fields," Number Theory (Noordwijkerhout, 1983), Lectures Notes in Math., vol. 1068, pp. 33-62, Springer-Verlag. Berlin and New York, 1984. MR 756082 (85j:11144)
  • [4] H. Cohen & J. Martinet, "Étude heuristique des groupes de classes." (In preparation.)
  • [5] H. Eisenbeis, G. Frey & B. Ommerborn, "Computation of the 2-rank of pure cubic fields," Math. Comp., v. 32, 1978, pp. 559-569. MR 0480416 (58:579)
  • [6] V. Ennola, S. Mäki & R. Turunen, "On real cyclic sextic fields," Math. Comp., v. 45, 1985, pp. 591-611. MR 804948 (86m:11084)
  • [7] V. Ennola & R. Turunen, "On totally real cubic fields," Math. Comp., v. 44, 1985, pp. 495-518. MR 777281 (86e:11100)
  • [8] V. Ennola & R. Turunen, "On cyclic cubic fields," Math. Comp. v. 45, 1985, pp. 585-589. MR 804947 (86m:11085)
  • [9] M.-N. Gras, "Méthodes et algorithmes pour le calcul numérique du nombre de classes et des unités des extensions cubiques cycliques de Q," J. Reine Angew. Math., v. 277, 1975, pp. 89-116. MR 0389845 (52:10675)
  • [10] M.-N. Gras, "Classes et unités des extensions cycliques réelles de degré 4 de Q," Ann. Inst. Fourier (Grenoble), v. 29, 1979, pp. 107-124. MR 526779 (81f:12003)
  • [11] C. Hooley, "On the Pellian equation and the class number of indefinite binary quadratic forms," J. Reine Angew. Math., v. 353, 1984, pp. 98-131. MR 765829 (86d:11032)
  • [12] S. Mäki, The Determination of Units in Real Cyclic Sextic Fields, Lecture Notes in Math., vol. 797, Springer-Verlag, Berlin and New York, 1980. MR 584794 (82a:12004)
  • [13] D. Shanks & H. C. Williams, "A note on class-number one in pure cubic fields," Math. Comp., v. 33, 1979, pp. 1317-1320. MR 537977 (80g:12002)
  • [14] C. P. Schnorr, Personal communication.
  • [15] M. Tennenhouse & H. C. Williams, "A note on class-number one in certain real quadratic and pure cubic fields," Math. Comp., v. 46, 1986, pp. 333-336. MR 815853 (87b:11127)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1987-0866103-4
Keywords: Class group, class number, number field, zeta function
Article copyright: © Copyright 1987 American Mathematical Society

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