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), 223-231

MSC:
Primary 11R16; Secondary 11R29, 11Y40

DOI:
https://doi.org/10.1090/S0025-5718-1985-0790655-4

Corrigendum:
Math. Comp. **50** (1988), 655-657.

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.

**[1]**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 2-rank of pure cubic fields,"*Math. Comp.*, v. 32, 1978, pp. 559-569. MR**0480416 (58:579)****[3]**V. Ennola & R. Turunen, "On totally real cubic fields,"*Math. Comp.*, v. 44, 1985, pp. 495-518. MR**777281 (86e:11100)****[4]**D. E. Knuth,*The Art of Computer Programming*. Vol. II:*Seminumerical Algorithms*, 2nd ed., Addison-Wesley, 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. 409-464. 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. 271-296. MR**553223 (81b:12013)****[7]**R. S. Lehman, "Factoring large integers,"*Math. Comp.*, v. 28, 1974, pp. 637-646. 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. 123-150. MR**697260 (86g:11080)****[9]**J. Oesterlé, "Versions effectives du théorème de Chebotarev sous l'hypothèse de Riemann généralisée,"*Astérisque*, v. 61, 1979, pp. 165-167.**[10]**R. Schoof, "Quadratic fields and factorization,"*Computational Methods in Number Theory*. Part II, Math. Centrum Tracts, No. 155, Amsterdam, 1983, pp. 235-286. 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. 415-440. 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. 217-224. 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. 235-286. MR**701638 (84m:12010)****[16]**H. C. Williams, "Continued fractions and number-theoretic 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. 285-306. MR**0389730 (52:10561)**

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DOI:
https://doi.org/10.1090/S0025-5718-1985-0790655-4

Article copyright:
© Copyright 1985
American Mathematical Society