Computation of real quadratic fields with class number one

Stephens, A. J.; Williams, H. C.

doi:10.1090/S0025-5718-1988-0958644-7

Computation of real quadratic fields with class number one
HTML articles powered by AMS MathViewer

by A. J. Stephens and H. C. Williams PDF

Math. Comp. 51 (1988), 809-824 Request permission

Abstract:

A rapid method for determining whether the real quadratic field $\mathcal {K} = \mathcal {Q}(\sqrt D )$ has class number one is described. The method makes use of the infrastructure idea of Shanks to determine the regulator of $\mathcal {K}$ and then uses the Generalized Riemann Hypothesis to rapidly estimate $L(1,\chi )$ to the accuracy needed for determining whether or not the class number of $\mathcal {K}$ is one. The results of running this algorithm on a computer for all prime values of D up to ${10^9}$ are also presented, together with further results on runs on intervals of size ${10^7}$ starting at ${10^i} (i = 9,10, \ldots ,16)$.

References

H. Cohen and H. W. Lenstra Jr., Heuristics on class groups of number fields, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983) Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 33–62. MR 756082, DOI 10.1007/BFb0099440
Gary Cornell and Lawrence C. Washington, Class numbers of cyclotomic fields, J. Number Theory 21 (1985), no. 3, 260–274. MR 814005, DOI 10.1016/0022-314X(85)90055-1
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

Astérisque

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, The infrastructure of a real quadratic field and its applications, Proceedings of the Number Theory Conference (Univ. Colorado, Boulder, Colo., 1972) Univ. Colorado, Boulder, Colo., 1972, pp. 217–224. MR 0389842
A. J. Stephens and H. C. Williams, Some computational results on a problem concerning powerful numbers, Math. Comp. 50 (1988), no. 182, 619–632. MR 929558, DOI 10.1090/S0025-5718-1988-0929558-3
A. J. Stephens and H. C. Williams, Some computational results on a problem of Eisenstein, Théorie des nombres (Quebec, PQ, 1987) de Gruyter, Berlin, 1989, pp. 869–886. MR 1024611
M. Tennenhouse and H. C. Williams, A note on class-number one in certain real quadratic and pure cubic fields, Math. Comp. 46 (1986), no. 173, 333–336. MR 815853, DOI 10.1090/S0025-5718-1986-0815853-3
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 and M. C. Wunderlich, On the parallel generation of the residues for the continued fraction factoring algorithm, Math. Comp. 48 (1987), no. 177, 405–423. MR 866124, DOI 10.1090/S0025-5718-1987-0866124-1