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The fundamental unit and bounds for class numbers of real quadratic fields

Published online by Cambridge University Press:  22 January 2016

Hideo Yokoi
Hideo Yokoi*
Affiliation:
Department of Mathematics, College of General Education Nagoya University, Chikusa-ku, Nagoya 464-01, Japan
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Although class number one problem for imaginary quadratic fields was solved in 1966 by A. Baker [3] and by H. M. Stark [25] independently, the problem for real quadratic fields remains still unsettled. However, since papers by Ankeny–Chowla–Hasse [2] and H. Hasse [9], many papers concerning this problem or giving estimate for class numbers of real quadratic fields from below have appeared. There are three methods used there, namely the first is related with quadratic diophantine equations ([2], [9], [27, 28, 29, 31], [17]), and the second is related with continued fraction expantions ([8], [4], [16], [14], [18]).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 124 , December 1991 , pp. 181 - 197
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1991

References

[ 1 ] Ankeny, N. C., Artin, E. and Chowla, S., The class-number of real quadratic number fields, Annals of Math., 56 (1952), 479–493.Google Scholar
[ 2 ] Ankeny, N. C., Chowla, S. and Hasse, H., On the class number of the real subfield of a cyclotomic field, J. Reine Angew. Math., 217 (1965), 217–220.Google Scholar
[ 3 ] Baker, A., Linear forms in the logarithms of algebraic numbers, I, Mathematika, 13 (1966), 204–216.Google Scholar
[ 4 ] Azuhata, T., On the fundamental units and the class numbers of real quadratic fields, II, Tokyo J. Math., 10–2 (1987), 259–270.Google Scholar
[ 5 ] Burgess, D. A., Estimating Lx(l), Det Kongelige Norske Videnskabers selskabs Forhandlinger, 39 (1966), 101–108.Google Scholar
[ 6 ] Chowla, S., Leopoldt’s criterion for real quadratic fields with class number 1, Abh. Math. Sem. Univ. Hamburg, 35 (1970), 32.Google Scholar
[ 7 ] Degert, G., Über die Bestimmung der Grundeinheit gewisser reell-quadratischer Zahlkorper, Abh. Math. Sem. Univ. Hamburg, 22 (1958), 92–97.Google Scholar
[ 8 ] Hendy, M. D., Applications of a continued fraction algorithm to some class number problems, Math. Comp., 28 (1974), 267–277.Google Scholar
[ 9 ] Hasse, H., Über mehrklassige, aber eingeschlechtige reell-quadratische Zahlkorper, Elemente der Mathematik, 20 (1965), 49–59.Google Scholar
[10] Hoffstein, J., On the Siegel-Tatuzawa theorem, Acta Arith., 38 (1980), 167–174.Google Scholar
[11] Hua, L.-K., On the least solution of Pell’s equation, Bull. Amer. Math. Soc, 48 (1942), 731–735.Google Scholar
[12] Kim, H. K., Leu, M.-G. and Ono, T., On two conjectures on real quadratic fields, Proc. Japan Acad., 63 (1987), 222–224.Google Scholar
[13] Kim, H. K., A conjecture of S. Chowla and related topics in analytic number theory, Thesis, Johns Hopkins University (1988).Google Scholar
[14] Koch, F. H.-, Quadratische Ordnungen mit grossen Klassenzahl, J. Number Theory, 34 (1990), 82–94.Google Scholar
[15] Leopoldt, H. W., Über Klassenzahl primteiler reeller abelscher Zahlkorper als Primteiler verallgemeinerter Bernoullische Zahlen, Abh. Math. Sem. Univ. Hamburg, 23 (1959), 36–47.Google Scholar
[16] Louboutin, S., Continued fractions and real quadratic fields, J. Number Theory, 30 (1988), 167–176.Google Scholar
[17] Mollin, R. A., Lower bounds for class numbers of real quadratic fields, Proc. A. M. S., 6 (1986), 545–550.Google Scholar
[18] Mollin, R. A., Class numbers bounded below by the divisor function, C. R. Math. Rep. Acad. Sci. Canada, 12 (1990), 119–124.Google Scholar
[19] Mollin, R. A. and Williams, H. C., A conjecture of S. Chowla via the generalized Riemann hypothesis, Proc. A. M. S., 102 (1988), 794–796.Google Scholar
[20] Mollin, R. A. and Williams, H. C., Solution of the class number one problem for real quadratic fields of extended Richaud-Degert type (with one possible exception), Proc. 1st Conf. Canadian Number Theory Assoc, April 18–27, 1988, Banff, Canada (1990), 417–425.Google Scholar
[21] Mollin, R. A. and Williams, H. C., Solution of a problem of Yokoi, Proc. Japan Acad., 66 (1990), 141–145.Google Scholar
[22] Nakahara, T., On the fundamental units and an estimate of the class number of real quadratic fields, Rep. Saga Univ. Math., 2 (1974), 1–13.Google Scholar
[23] Newman, M., Bounds for class numbers, Proc. Sympo. pure Math., VIII (1965),70–77.Google Scholar
[24] Richaud, C., Sur la resolution des equations x2 — Ay2 =; ±1 , Atti Accad. Pontif. Nuovi Lincei (1866), 177–182.Google Scholar
[25] Stark, H. M., A complete determination of the complex quadratic fields of class number one, Michigan Math. J., 14 (1967), 1–27.Google Scholar
[26] Tatuzawa, T., On a theorem of Siegal, Japan J. Math., 21 (1951), 163–178.CrossRefGoogle Scholar
[27] Yokoi, H., On real quadratic fields containing units with norm —1, Nagoya Math. J., 33 (1968), 139–152.Google Scholar
[28] Yokoi, H., On the diophantine equation x2 — py2 = ± 4q and the class number of real subfields of a cyclotomic field, Nagoya Math. J., 91 (1983), 151–161.Google Scholar
[29] Yokoi, H., Class-number one problem for certain kind of real quadratic fields, Proc. Int. Conf. on Class Numbers and Fundamental Units of Algebraic Number Fields, June 24–28, 1986, Katata Japan, 125–137.Google Scholar
[30] Yokoi, H., Class number one problem for real quadratic fields (The conjecture of Gauss), Proc. Japan Acad., 64–2 (1988), 53–55.Google Scholar
[31] Yokoi, H., Some relations among new invariants of prime number p congruent to 1 mod 4, Advances in pure Math., 13 (1988), 493–501.Google Scholar
[32] Yokoi, H., New invariants of real quadratic fields, Proc. 1st Conf. Canadian Number Theory Assoc, April 18–27, 1988, Banff, Canada (1990), 635–639.Google Scholar
[33] Yokoi, H., Bounds for fundamental units and class numbers of real quadratic fields with prime discriminant, Dept. of Math., Coll. Gen. Educ, Nagoya Univ., Prep. ser., 1989, No. 2.Google Scholar
[34] Yokoi, H., The fundamental unit and class number one problem of real quadratic fields with prime discriminant, Nagoya Math. J., 120 (1990), 51–59.CrossRefGoogle Scholar