Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-08T02:11:02.033Z Has data issue: false hasContentIssue false
  • English
  • Français

New invariants and class number problem in real quadratic fields

Published online by Cambridge University Press:  22 January 2016

Hideo Yokoi
Hideo Yokoi*
Affiliation:
Graduate School of Human Informatics Nagoya University Chikusa-ku, Nagoya 464-01Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In recent papers [10, 11, 12, 13, 14], we defined some new ρ-invariants for any rational prime ρ congruent to 1 mod 4 and D-invariants for any positive square-free integer D such that the fundamental unit εD of real quadratic field Q(√D) satisfies NεD = –1, and studied relationships among these new invariants and already known invariants.

One of our main purposes in this paper is to generalize these D-invariants to invariants valid for all square-free positive integers containing D with NεD = 1. Another is to provide an improvement of the theorem in [14] related closely to class number one problem of real quadratic fields. Namely, we provide, in a sense, a most appreciable estimation of the fundamental unit to be able to apply, as usual (cf. [3, 4, 5, 9, 12, 13]), Tatuzawa’s lower bound of L(l, XD) (Cf[7]) for estimating the class number of Q(√D) from below by using Dirichlet’s classical class number formula.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 132 , December 1993 , pp. 175 - 197
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1993

References

[1] Degert, G., Über die Bestimmung der Grundeinheit gewisser reell-quadratischer Zahlkörper, Abh. Math. Sem. Univ. Hamburg, 22 (1958), 92–97.CrossRefGoogle Scholar
[2] Hasse, H., Über mehrklassige, aber eingeschlechtige reell-quadratische Zahlkörper, Elemente der Mathematik, 20 (1965), 49–59.Google Scholar
[3] Katayama, Shinichi and Katayama, Shigeru, On bounds for fundamental units of real quadratic fields, to appear in J. Number Theory.Google Scholar
[4] Mollin, R. A. and Williams, H. C., Solution of the class number one problem for realquadratic fields of extended Richaud-Degert type (with one possible exception). Proc. 1st Conf. Canadian Number Theory Assoc, April 18–27, 1988, Banff, Canada, pp. 417–425 (1990).Google Scholar
[5] 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
[6] Richaud, C., Sur la resolution des equations x1 — Ay2 = ± 1, Atti Accad. Pontif. Nuovi Lincei (1866), 177–182.Google Scholar
[7] Tatuzawa, T., On a theorem of Siegel, Japan J. Math., 21 (1951), 163–178.Google Scholar
[8] Yokoi, H., On real quadratic fields containing units with norm — 1, Nagoya Math. J., 33 (1968), 139–152.CrossRefGoogle Scholar
[9] Yokoi, H., Class number one problem for real quadratic fields (The conjecture of Gauss), Proc. Japan Acad., 64-2 (1988), 53–55.Google Scholar
[10] Yokoi, H., Some relations among new invariants of prime number p congruent to 1 mod 4, Advances in Pure Math. 13 (1988), 493–501.CrossRefGoogle Scholar
[11] Yokoi, H., New invariants of real quadratic fields, Proc. 1 Conf. Canadian Number Theory Assoc, April 18-27, 1988, Banff, Canada, pp. 635–639 (1990).Google Scholar
[12] 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
[13] Yokoi, H., The fundamental unit and bounds for class numbers of real quadratic fields, Nagoya Math. J., 124 (1991), 181–197.CrossRefGoogle Scholar
[14] Yokoi, H., A note on class number one problem for real quadratic fields, Proc. Japan Acad., 69 (1993), 22–26.Google Scholar