A note on class-number one in certain real quadratic and pure cubic fields

Tennenhouse, M.; Williams, H. C.

doi:10.1090/S0025-5718-1986-0815853-3

A note on class-number one in certain real quadratic and pure cubic fields

Authors: M. Tennenhouse and H. C. Williams
Journal: Math. Comp. 46 (1986), 333-336
MSC: Primary 11Y40; Secondary 11R11, 11R16
DOI: https://doi.org/10.1090/S0025-5718-1986-0815853-3
MathSciNet review: 815853
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Abstract: Let p be any odd prime and let be the class number of the real quadratic field $\mathcal{Q}(\sqrt p )$ . The results of a computer run to determine the density of the field $\mathcal{Q}(\sqrt p )$ with and $p < {10^8}$ are presented. Similar results are given for pure cubic fields $\mathcal{Q}(\sqrt[3]{p})$ with $p < {10^6}$ .

[1] Henri Cohen, Sur la distribution asymptotique des groupes de classes, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 5, 245–247 (French, with English summary). MR 693784
[2] 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, https://doi.org/10.1090/S0025-5718-1978-0480416-4
[3] Taira Honda, Pure cubic fields whose class numbers are multiples of three, J. Number Theory 3 (1971), 7–12. MR 292795, https://doi.org/10.1016/0022-314X(71)90045-X
[4] S. Kuroda, "Table of class numbers for quadratic fields $Q(\sqrt p )$ , $p \leqslant 2776817$ ," Math. Comp., v. 29, 1975, pp. 335-336, UMT File.
[5] Richard B. Lakein, Computation of the ideal class group of certain complex quartic fields. II, Math. Comp. 29 (1975), 137–144. MR 444605, https://doi.org/10.1090/S0025-5718-1975-0444605-4
[6] 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
[7] 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
[8] 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
[9] H. C. Williams, Improving the speed of calculating the regulator of certain pure cubic fields, Math. Comp. 35 (1980), no. 152, 1423–1434. MR 583520, https://doi.org/10.1090/S0025-5718-1980-0583520-4
[10] 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, https://doi.org/10.1216/RMJ-1985-15-2-621
[11] H. C. Williams and J. Broere, A computational technique for evaluating 𝐿(1,𝜒) and the class number of a real quadratic field, Math. Comp. 30 (1976), no. 136, 887–893. MR 414522, https://doi.org/10.1090/S0025-5718-1976-0414522-5
[12] 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, https://doi.org/10.1090/S0025-5718-1983-0701638-2
[13] H. C. Williams and Daniel Shanks, A note on class-number one in pure cubic fields, Math. Comp. 33 (1979), no. 148, 1317–1320. MR 537977, https://doi.org/10.1090/S0025-5718-1979-0537977-7