A note on classnumber one in certain real quadratic and pure cubic fields
Authors:
M. Tennenhouse and H. C. Williams
Journal:
Math. Comp. 46 (1986), 333336
MSC:
Primary 11Y40; Secondary 11R11, 11R16
MathSciNet review:
815853
Fulltext PDF Free Access
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Abstract: Let p be any odd prime and let be the class number of the real quadratic field . The results of a computer run to determine the density of the field with and are presented. Similar results are given for pure cubic fields with .
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 [1]
 H. Cohen, "Sur la distribution asymptotique des groupes de classes," C. R. Acad. Sci. Paris Sér. I Math., v. 296, 1983, pp. 245246. MR 693784 (84c:12001)
 [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]
 T. Honda, "Pure cubic fields whose class numbers are multiples of three," J. Number Theory, v. 3, 1971, pp. 712. MR 0292795 (45:1877)
 [4]
 S. Kuroda, "Table of class numbers for quadratic fields , ," Math. Comp., v. 29, 1975, pp. 335336, UMT File.
 [5]
 R. B. Lakein, "Computation of the ideal class group of certain complex quartic fields, II," Math. Comp., v. 29, 1975, pp. 137144. MR 0444605 (56:2955)
 [6]
 H. W. Lenstra, Jr., On the Calculation of Regulators and Class Numbers of Quadratic Fields, London Math. Soc. Lecture Note Series, no. 56, 1982, pp. 123150. MR 697260 (86g:11080)
 [7]
 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)
 [8]
 D. Shanks, Class Number, A Theory of Factorization, and Genera, Proc. Sympos. Pure Math., Vol. 20 (1969 Institute on Number Theory), Amer. Math. Soc., Providence, R. I., 1971, pp. 415440. MR 0316385 (47:4932)
 [9]
 H. C. Williams, "Improving the speed of calculating the regulator of certain pure cubic fields," Math. Comp., v. 35, 1980, pp. 14231434. MR 583520 (82a:12003)
 [10]
 H. C. Williams, "Continued fractions and numbertheoretic computations" (Proc. Number Theory Conf. Edmonton 1983), Rocky Mountain J. Math., v. 15, 1985, pp. 621655. MR 823273 (87h:11129)
 [11]
 H. C. Williams & J. Broere, "A computational technique for evaluating and the class number of a real quadratic field," Math. Comp., v. 30, 1976, pp. 887893. MR 0414522 (54:2623)
 [12]
 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)
 [13]
 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/S00255718198608158533
PII:
S 00255718(1986)08158533
Article copyright:
© Copyright 1986 American Mathematical Society
