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On prime valued polynomials and class numbers of real quadratic fields

Published online by Cambridge University Press:  22 January 2016

R.A. Mollin  and
H.C. Williams
R.A. Mollin
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada, T2N 1N4
H.C. Williams
Affiliation:
Department of Computer Science, University of Manitoba, Winnipeg, Manitoba, Canada, RST 2N2
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Gauss conjectured that there are infinitely many real quadratic fields with class number one. Today this is still an open problem. Moreover, as Dorian Goldfeld, one of the recipients of the 1987 Cole prize in number theory (for his work on another problem going back to Gauss) recently stated in his acceptance of the award: “This problem appears quite intractible at the moment.” However there has recently been a search for conditions which are tantamount to class number one for real quadratic fields. This may be viewed as an effort to shift the focus of the problem in order to understand more clearly the inherent difficulties, and to reveal some other beautiful interrelationships.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 112 , December 1988 , pp. 143 - 151
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1988

References

[1] Azuhata, T., On the fundamental units and the class numbers of real quadratic fields, Nagoya Math. J., 95 (1984), 125–135.Google Scholar
[2] Baker, A., Linear forms in the logarithms of algebraic numbers, Mathematika, 13 (1966), 204–216.CrossRefGoogle Scholar
[3] Chowla, S. and Friedlander, J., Class numbers and quadratic residues, Glasgow Math. J., 17 (1976), 47–52.CrossRefGoogle Scholar
[4] Degert, G., Uber die bestimmung der grundeinheit gewisser reellquadratischen zahlkorper, Abh. Math. Sem. Univ. Hamburg, 22 (1958), 92–97.Google Scholar
[5] Goldfeld, D., Gauss’ class number problems for imaginary quadratic fields, Bull. Amer. Math. Soc., 13 (1985), 23–37.Google Scholar
[6] Kutsuna, M., On a criterion for the class number of a quadratic number field to be one, Nagoya Math. J., 79 (1980), 123–129.Google Scholar
[7] Louboutin, S., Critères des principauté et minoration des nombres de classes l’ideaux des corps quadratiques réells à l’aide de la théorie des fractions continues, preprint.Google Scholar
[8] Mollin, R. A., Necessary and sufficient conditions for the class number of a real quadratic field to be one, and a conjecture of S. Chowla, Proceedings Amer. Math. Soc., 102 (1988), 17–21.Google Scholar
[9] Mollin, R. A., Class number one criteria for real quadratic fields I, Proceedings Japan Acad., Series A, 83 (1987), 121–125.Google Scholar
[10] Mollin, R. A., On the insolubility of a class of diophantine equations and the nontriviality of the class numbers of related real quadratic fields of Richaud-Degert type, Nagoya Math. J., 105 (1987), 39–47.Google Scholar
[11] Mollin, R. A., Diophantine equations and class numbers, J. Number Theory, 24 (1986), 7–19.CrossRefGoogle Scholar
[12] Mollin, R. A. and Williams, H. C., A conjecture of S. Chowla via the generalized Riemann hypothesis, Proceedings Amer. Math. Soc., 102 (1988), 794–796.Google Scholar
[13] Rabinowitsch, G., Eindentigkeit der zerlegung in primzahlfaktoren in quadratischen zahlkörpern, Proc. Fifth Internat. Congress Math. (Cambridge) Vol. 1 (1913), 418–421.Google Scholar
[14] Mollin, R. A., Eindeutigkeit der zerlegung in primzahlfaktoren in quadratischen zahlkörpern, J. reine angew. Math., 142 (1913), 153–164.Google Scholar
[15] Richaud, C., Sur la résolution des équations x2 – Ay2 = ± 1, Atti. Acad. Pontif. Nuovi. Lincei (1866), 177–182.Google Scholar
[16] Sasaki, R., Generalized ono invariant and Rabinovitch’s theorem for real quadratic fields, preprint.Google Scholar
[17] Stark, H., A complete determination of the complex quadratic fields of class-number one, Michigan Math. J., 14 (1967), 1–27.Google Scholar
[18] Yokoi, H., Class-number one problem for certain kind of real quadratic fields, Proc. Int. Conf. on class numbers and fundamental units, June 1986, Katata Japan, 125–137.Google Scholar