Computation of the class number and class group of a complex cubic field
Authors:
G. Dueck and H. C. Williams
Journal:
Math. Comp. 45 (1985), 223231
MSC:
Primary 11R16; Secondary 11R29, 11Y40
Corrigendum:
Math. Comp. 50 (1988), 655657.
MathSciNet review:
790655
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Abstract: Let h and G be, respectively, the class number and the class group of a complex cubic field of discriminant . A method is described which makes use of recent ideas of Lenstra and Schoof to develop fast algorithms for finding h and G. Under certain Riemann hypotheses it is shown that these algorithms will compute h in elementary operations and G in elementary operations. Finally, the results of running some computer programs to determine h and G for all pure cubic fields , with , are summarized.
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 B. N. Delone & D. K. Faddeev, The Theory of Irrationalities of the Third Degree, Transl. Math. Monographs, Vol. 10, Amer. Math. Soc., Providence, R. I., 1964. MR 0160744 (28:3955)
 [2]
 H. Eisenbeis, G. Frey & B. Ommerborn, "Computation of the 2rank of pure cubic fields," Math. Comp., v. 32, 1978, pp. 559569. MR 0480416 (58:579)
 [3]
 V. Ennola & R. Turunen, "On totally real cubic fields," Math. Comp., v. 44, 1985, pp. 495518. MR 777281 (86e:11100)
 [4]
 D. E. Knuth, The Art of Computer Programming. Vol. II: Seminumerical Algorithms, 2nd ed., AddisonWesley, Reading, Mass., 1981. MR 633878 (83i:68003)
 [5]
 J. C. Lagarias & A. M. Odlyzko, "Effective versions of the Chebotarev density theorem," Algebraic Number Fields (A. Fröhlich, ed.), Academic Press, London, 1977, pp. 409464. MR 0447191 (56:5506)
 [6]
 J. C. Lagarias, H. L. Montgomery & A. M. Odlyzko, "A bound for the least prime ideal in the Chebotarev density theorem," Invent. Math., v. 54, 1979, pp. 271296. MR 553223 (81b:12013)
 [7]
 R. S. Lehman, "Factoring large integers," Math. Comp., v. 28, 1974, pp. 637646. MR 0340163 (49:4919)
 [8]
 H. W. Lenstra, Jr., On the Calculation of Regulators and Class Numbers of Quadratic Fields, London Math. Soc. Lecture Note Ser., Vol. 56, 1982, pp. 123150. MR 697260 (86g:11080)
 [9]
 J. Oesterlé, "Versions effectives du théorème de Chebotarev sous l'hypothèse de Riemann généralisée," Astérisque, v. 61, 1979, pp. 165167.
 [10]
 R. Schoof, "Quadratic fields and factorization," Computational Methods in Number Theory. Part II, Math. Centrum Tracts, No. 155, Amsterdam, 1983, pp. 235286. MR 702519 (85g:11118b)
 [11]
 D. Shanks, Class Number, A Theory of Factorization and Genera, Proc. Sympos. Pure Math., Vol. 20, Amer. Math. Soc., Providence, R. I., 1971, pp. 415440. MR 0316385 (47:4932)
 [12]
 D. Shanks, The Infrastructure of a Real Quadratic Field and Its Applications, Proc. 1972 Number Theory Conf. (Boulder, 1972), pp. 217224. MR 0389842 (52:10672)
 [13]
 G. F. Voronoi, Concerning Algebraic Integers Derivable from a Root of an Equation of the Third Degree, Master's Thesis, St. Petersburg, 1894. (Russian)
 [14]
 G. F. Voronoi, On a Generalization of the Algorithm of Continued Fractions, Doctoral Dissertation, Warsaw, 1896. (Russian)
 [15]
 H. C. Williams, G. W. Dueck & B. K. Schmid, "A rapid method of evaluating the regulator and class number of a pure cubic field," Math. Comp., v. 41, 1983, pp. 235286. MR 701638 (84m:12010)
 [16]
 H. C. Williams, "Continued fractions and numbertheoretic computations," Proc. Number Theory Conf. (Edmonton, 1983); Rocky Mountain J. Math. (To appear.) MR 823273 (87h:11129)
 [17]
 H. C. Williams & C. R. Zarnke, "Some algorithms for solving a cubic congruence modulo p," Utilitas Math., v. 6, 1974, pp. 285306. MR 0389730 (52:10561)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198507906554
PII:
S 00255718(1985)07906554
Article copyright:
© Copyright 1985
American Mathematical Society
