A rapid method of evaluating the regulator and class number of a pure cubic field
Authors:
H. C. Williams, G. W. Dueck and B. K. Schmid
Journal:
Math. Comp. 41 (1983), 235286
MSC:
Primary 12A50; Secondary 1204, 12A30, 12A45
MathSciNet review:
701638
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Abstract: Let be the algebraic number field formed by adjoining to the rationals . Let R and h be, respectively, the regulator and class number of . Shanks has described a method of evaluating R for , where D is a positive integer. His technique improved the speed of the usual continued fraction algorithm for finding R by allowing one to proceed almost directly from the nth to the mth step, where m is approximately 2n, in the continued fraction expansion of . This paper shows how Shanks' idea can be extended to the Voronoi algorithm, which is used to find R in cubic fields of negative discriminant. It also discusses at length an algorithm for finding R and h for pure cubic fields , D an integer. Under a certain generalized Riemann Hypothesis the ideas developed here will provide a new method which will find R and h in operations. When h is small, this is an improvement over the operations required by Voronoi's algorithm to find R. For example, with , it required only 5 minutes for an AMDAHL 470/V7 computer to find that and . This same calculation would require about 8 days of computer time if it used only the standard Voronoi algorithm.
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 [1]
 P. Barrucand, H. C. Williams & L. Baniuk, "A computational technique for determining the class number of a pure cubic field," Math. Comp., v. 30, 1976, pp. 312323. MR 0392913 (52:13726)
 [2]
 W. E. H. Berwick, "The classification of ideal numbers that depend on a cubic irrationality," Proc. London Math. Soc., v. 12, 1913, pp. 393429.
 [3]
 W. E. H. Berwick, "The arithmetic of quadratic number fields," Math. Gazette, v. 14, 1928, pp. 111.
 [4]
 J. W. S. Cassels, "The rational solutions of the diophantine equation ," Acta Math., v. 82, 1950, pp. 243273. MR 0035782 (12:11a)
 [5]
 G. Cornell & L. Washington, "Class numbers of cyclotomic fields," J. Number Theory. (To appear.) MR 814005 (87d:11079)
 [6]
 B. N. Delone & D. K. Faddeev, The Theory of Irrationalities of the Third Degree, Transl. Math. Mono., vol. 10, Amer. Math. Soc., Providence, R.I., 1964. MR 0160744 (28:3955)
 [7]
 J. C. Lagarias & A. M. Odlyzko, "Effective versions of the Chebotarev density theorem," Algebraic Number Fields, Academic Press, New York, 1977, pp. 409464. MR 0447191 (56:5506)
 [8]
 E. Landau, Vorlesungen über Zahlentheorie, Vol. II, Chelsea, New York, 1955.
 [9]
 H. W. Lenstra, Jr., On the Calculation of Regulators and Class Numbers of Quadratic Fields, Report 8008, Mathematisch Instituut, Amsterdam, 1980.
 [10]
 J. Oesterlé, "Versions effectives du théorème de Chebotarev sous l'hypothése de Riemann généralisée," Astérisque, v. 61, 1979, pp. 165167.
 [11]
 R. J. Schoof, Quadratic fields and factorization, Studie Week Getaltheorie en Computers, Math. Centrum, Amsterdam, 1980, pp. 165206. MR 702519 (85g:11118b)
 [12]
 D. Shanks, The Infrastructure of a Real Quadratic Field and its Applications, Proc. 1972 Number Theory Conference, Boulder, 1972, pp. 217224. MR 0389842 (52:10672)
 [13]
 D. Shanks, "A survey of quadratic, cubic and quartic algebraic number fields (from a computational point of view)," Congressus Numerantium, v. 17, 1976, pp. 1540. MR 0453691 (56:11951)
 [14]
 G. F. Voronoi, Concerning Algebraic Integers Derivable from a Root of an Equation of the Third Degree, Master's Thesis, St. Petersburg, 1894. (Russian)
 [15]
 G. F. Voronoi, On a Generalization of the Algorithm of Continued Fractions, Doctoral Dissertation, Warsaw, 1896. (Russian)
 [16]
 H. C. Williams, "Improving the speed of calculating the regulator of certain pure cubic fields," Math. Comp., v. 34, 1980, pp. 14231434. MR 583520 (82a:12003)
 [17]
 H. C. Williams, "Some results concerning Voronoi's Continued Fraction over ." Math. Comp., v. 36, 1981, pp. 631652. MR 606521 (82j:12011)
 [18]
 H. C. Williams, G. Cormack & E. Seah, "Calculation of the regulator of a pure cubic field," Math. Comp., v. 34, 1980, pp. 567611. MR 559205 (81d:12003)
 [19]
 H. C. Williams & D. Shanks, "A note on classnumber one in pure cubic fields," Math. Comp., v. 33, 1979, pp. 13171320. MR 537977 (80g:12002)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198307016382
PII:
S 00255718(1983)07016382
Article copyright:
© Copyright 1983 American Mathematical Society
