A note on class-number one in certain real quadratic and pure cubic fields
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- by M. Tennenhouse and H. C. Williams PDF
- Math. Comp. 46 (1986), 333-336 Request permission
Abstract:
Let p be any odd prime and let $h(p)$ 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 $h(p) = 1$ and $p < {10^8}$ are presented. Similar results are given for pure cubic fields $\mathcal {Q}(\sqrt [3]{p})$ with $p < {10^6}$.References
- 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
- 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
- Taira Honda, Pure cubic fields whose class numbers are multiples of three, J. Number Theory 3 (1971), 7–12. MR 292795, DOI 10.1016/0022-314X(71)90045-X S. Kuroda, "Table of class numbers $h(p) > 1$ for quadratic fields $Q(\sqrt p )$, $p \leqslant 2776817$," Math. Comp., v. 29, 1975, pp. 335-336, UMT File.
- Richard B. Lakein, Computation of the ideal class group of certain complex quartic fields. II, Math. Comp. 29 (1975), 137–144. MR 444605, DOI 10.1090/S0025-5718-1975-0444605-4
- 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
- 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
- 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, DOI 10.1090/S0025-5718-1980-0583520-4
- 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 J. Broere, A computational technique for evaluating $L(1,\chi )$ and the class number of a real quadratic field, Math. Comp. 30 (1976), no. 136, 887–893. MR 414522, DOI 10.1090/S0025-5718-1976-0414522-5
- 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
- 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, DOI 10.1090/S0025-5718-1979-0537977-7
Additional Information
- © Copyright 1986 American Mathematical Society
- 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