## Computation of the class number and class group of a complex cubic field

HTML articles powered by AMS MathViewer

- by G. Dueck and H. C. Williams PDF
- Math. Comp.
**45**(1985), 223-231 Request permission

Corrigendum: Math. Comp.

**50**(1988), 655-657.

## Abstract:

Let*h*and

*G*be, respectively, the class number and the class group of a complex cubic field of discriminant $\Delta$. 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 $O(|\Delta {|^{1/5 + \varepsilon }})$ elementary operations and

*G*in $O(|\Delta {|^{1/4 + \varepsilon }})$ elementary operations. Finally, the results of running some computer programs to determine

*h*and

*G*for all pure cubic fields $\mathcal {Q}(\sqrt [3]{D})$, with $2 \leqslant D < 30,000$, are summarized.

## References

- B. N. Delone and D. K. Faddeev,
*The theory of irrationalities of the third degree*, Translations of Mathematical Monographs, Vol. 10, American Mathematical Society, Providence, R.I., 1964. MR**0160744** - H. Eisenbeis, G. Frey, and B. Ommerborn,
*Computation of the $2$-rank of pure cubic fields*, Math. Comp.**32**(1978), no. 142, 559–569. MR**480416**, DOI 10.1090/S0025-5718-1978-0480416-4 - Veikko Ennola and Reino Turunen,
*On totally real cubic fields*, Math. Comp.**44**(1985), no. 170, 495–518. MR**777281**, DOI 10.1090/S0025-5718-1985-0777281-8 - Donald E. Knuth,
*The art of computer programming. Vol. 2*, 2nd ed., Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass., 1981. Seminumerical algorithms. MR**633878** - J. C. Lagarias and A. M. Odlyzko,
*Effective versions of the Chebotarev density theorem*, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 409–464. MR**0447191** - J. C. Lagarias, H. L. Montgomery, and A. M. Odlyzko,
*A bound for the least prime ideal in the Chebotarev density theorem*, Invent. Math.**54**(1979), no. 3, 271–296. MR**553223**, DOI 10.1007/BF01390234 - R. Sherman Lehman,
*Factoring large integers*, Math. Comp.**28**(1974), 637–646. MR**340163**, DOI 10.1090/S0025-5718-1974-0340163-2 - H. W. Lenstra Jr.,
*On the calculation of regulators and class numbers of quadratic fields*, Number theory days, 1980 (Exeter, 1980) London Math. Soc. Lecture Note Ser., vol. 56, Cambridge Univ. Press, Cambridge, 1982, pp. 123–150. MR**697260**
J. Oesterlé, "Versions effectives du théorème de Chebotarev sous l’hypothèse de Riemann généralisée," - H. Zantema,
*Class numbers and units*, Computational methods in number theory, Part II, Math. Centre Tracts, vol. 155, Math. Centrum, Amsterdam, 1982, pp. 213–234. MR**702518** - Daniel Shanks,
*Class number, a theory of factorization, and genera*, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence, R.I., 1971, pp. 415–440. MR**0316385** - Daniel Shanks,
*The infrastructure of a real quadratic field and its applications*, Proceedings of the 1972 Number Theory Conference (Univ. Colorado, Boulder, Colo.), Univ. Colorado, Boulder, Colo., 1972, pp. 217–224. MR**0389842**
G. F. Voronoi, - H. C. Williams, G. W. Dueck, and B. K. Schmid,
*A rapid method of evaluating the regulator and class number of a pure cubic field*, Math. Comp.**41**(1983), no. 163, 235–286. MR**701638**, DOI 10.1090/S0025-5718-1983-0701638-2 - Hugh C. Williams,
*Continued fractions and number-theoretic computations*, Rocky Mountain J. Math.**15**(1985), no. 2, 621–655. Number theory (Winnipeg, Man., 1983). MR**823273**, DOI 10.1216/RMJ-1985-15-2-621 - H. C. Williams and C. R. Zarnke,
*Some algorithms for solving a cubic congruence modulo $p$*, Utilitas Math.**6**(1974), 285–306. MR**389730**

*Astérisque*, v. 61, 1979, pp. 165-167.

*Concerning Algebraic Integers Derivable from a Root of an Equation of the Third Degree*, Master’s Thesis, St. Petersburg, 1894. (Russian) G. F. Voronoi,

*On a Generalization of the Algorithm of Continued Fractions*, Doctoral Dissertation, Warsaw, 1896. (Russian)

## Additional Information

- © Copyright 1985 American Mathematical Society
- Journal: Math. Comp.
**45**(1985), 223-231 - MSC: Primary 11R16; Secondary 11R29, 11Y40
- DOI: https://doi.org/10.1090/S0025-5718-1985-0790655-4
- MathSciNet review: 790655