Computation of real quadratic fields with class number one
Authors:
A. J. Stephens and H. C. Williams
Journal:
Math. Comp. 51 (1988), 809824
MSC:
Primary 11R11; Secondary 11R29, 11Y40
MathSciNet review:
958644
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Abstract 
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Abstract: A rapid method for determining whether the real quadratic field has class number one is described. The method makes use of the infrastructure idea of Shanks to determine the regulator of and then uses the Generalized Riemann Hypothesis to rapidly estimate to the accuracy needed for determining whether or not the class number of is one. The results of running this algorithm on a computer for all prime values of D up to are also presented, together with further results on runs on intervals of size starting at .
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 [1]
 H. Cohen & H. W. Lenstra, Jr., "Heuristics on class groups of number fields," Number Theory (Noordwijkerhout, 1983), Lecture Notes in Math., vol. 1068, SpringerVerlag, Berlin and New York, 1984, pp. 3362. MR 756082 (85j:11144)
 [2]
 G. Cornell & L. C. Washington, "Class numbers of cyclotomic fields," J. Number Theory, v. 21, 1985, pp. 260274. MR 814005 (87d:11079)
 [3]
 H. W. Lenstra, Jr., "On the calculation of regulators and class numbers of quadratic fields," London Math. Soc. Lecture Note Ser., v. 56, 1982, pp. 123150. MR 697260 (86g:11080)
 [4]
 J. Oesterlé, "Versions effectives du théorème de Chebotarev sous l'hypothèse de Riemann généralisée," Astérisque, v. 61, 1979, pp. 165167.
 [5]
 R. J. Schoof, "Quadratic fields and factorization," Computational Methods in Number Theory (H. W. Lenstra, Jr. and R. Tijdemann, eds.), Math. Centrum Tracts, Number 155, Part II, Amsterdam, 1983, pp. 235286. MR 702519 (85g:11118b)
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 D. Shanks, "The infrastructure of a real quadratic field and its applications," Proc. 1972 Number Theory Conference (Univ. Colorado, Boulder, 1972), pp. 217224, Univ. Colorado, Boulder, 1972. MR 0389842 (52:10672)
 [7]
 A. J. Stephens & H. C. Williams, "Some computational results on a problem concerning powerful numbers," Math. Comp., v. 50, 1988, pp. 619632. MR 929558 (89d:11091)
 [8]
 A. J. Stephens & H. C. Williams, "Some computational results on a problem of Eisenstein," Proc. International Number Theory Conf., Laval University, Québec, 1987. (To appear.) MR 1024611 (91c:11066)
 [9]
 M. Tennenhouse & H. C. Williams, "A note on classnumber one in certain real quadratic and pure cubic fields," Math. Comp., v. 46, 1986, pp. 333336. MR 815853 (87b:11127)
 [10]
 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)
 [11]
 H. C. Williams & M. C. Wunderlich, "On the parallel generation of the residues for the continued fraction factoring algorithm," Math. Comp., v. 48, 1987, pp. 405423. MR 866124 (88i:11099)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198809586447
PII:
S 00255718(1988)09586447
Article copyright:
© Copyright 1988
American Mathematical Society
