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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $p < {10^6}$.

- Henri Cohen,
*Sur la distribution asymptotique des groupes de classes*, C. R. Acad. Sci. Paris SÃŠr. I Math.**296**(1983), no. 5, 245â247 (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, 559â569. MR**480416**, DOI https://doi.org/10.1090/S0025-5718-1978-0480416-4 - Taira Honda,
*Pure cubic fields whose class numbers are multiples of three*, J. Number Theory**3**(1971), 7â12. MR**292795**, DOI https://doi.org/10.1016/0022-314X%2871%2990045-X
S. Kuroda, "Table of class numbers $h(p) > 1$ for quadratic fields $Q(\sqrt p )$, $p \leqslant 2776817$," - Richard B. Lakein,
*Computation of the ideal class group of certain complex quartic fields. II*, Math. Comp.**29**(1975), 137â144. MR**444605**, DOI https://doi.org/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. 123â150. 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. 213â234. 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. 415â440. MR**0316385** - H. C. Williams,
*Improving the speed of calculating the regulator of certain pure cubic fields*, Math. Comp.**35**(1980), no. 152, 1423â1434. MR**583520**, DOI https://doi.org/10.1090/S0025-5718-1980-0583520-4 - Hugh C. Williams,
*Continued fractions and number-theoretic computations*, Rocky Mountain J. Math.**15**(1985), no. 2, 621â655. Number theory (Winnipeg, Man., 1983). MR**823273**, DOI https://doi.org/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, 887â893. MR**414522**, DOI https://doi.org/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, 235â286. MR**701638**, DOI https://doi.org/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, 1317â1320. MR**537977**, DOI https://doi.org/10.1090/S0025-5718-1979-0537977-7

*Math. Comp.*, v. 29, 1975, pp. 335-336, UMT File.

Retrieve articles in *Mathematics of Computation*
with MSC:
11Y40,
11R11,
11R16

Retrieve articles in all journals with MSC: 11Y40, 11R11, 11R16

Additional Information

Article copyright:
© Copyright 1986
American Mathematical Society