On certain real quadratic fields with class number one. (English) Zbl 0963.11064

For a positive square-free integer \(D\), denote by \(\varepsilon_D= (t+u\sqrt{D})/2\) the fundamental unit \((>1)\) of the real quadratic field \(k= \mathbb Q(\sqrt{D})\). Then, the reviewer showed that there exist exactly 30 real quadratic fields \(\mathbb Q(\sqrt{p})\) of class number one satisfying \(\varepsilon_p< 2p\) with one more possible exception of prime discriminant \(p\) [Nagoya Math. J. 120, 51-59 (1990; Zbl 0701.11046)]. Since then, Katayama, Mollin-Williams and others have shown some analogous results on the class number one problem, and in this paper the authors found the following five real quadratic fields \(\mathbb Q(\sqrt{D})\) with class number one for \(1\leq u\leq 100\) with one possible exception by the help of a computer and using Kida’s UBASIC 86: \((u,D)= (40,57), (77,893), (78,19), (84,22), (85,1397)\).


11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
11R27 Units and factorization


Full Text: DOI


[1] Karaali, F., and \.Işcan, H.: Class number two problem for real quadratic fields with fundamental units with the positive norm. Proc. Japan Acad., 74A , 139-141 (1998). · Zbl 0940.11052
[2] Katayama, S. I., and Katayama, S.-G.:[3] A note on the problem of Yokoi. Proc. Japan Acad., 67A , 26-28 (1991). · Zbl 0736.11065
[3] Kida, Y.: UBASIC 86. Nihonhyoronsha, Tokyo (1988). · Zbl 0692.10002
[4] 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). Number Theory (ed. Mollin, R. A.). Walter de Gruyter, Berlin-New York, pp. 417-425 (1990). · Zbl 0696.12004
[5] Mollin, R. A., and Williams, H. C.: A complete generalization of Yokoi’s \(p\)-invariants. Colloq. Math., 63 , fasc. 2, 285-294 (1992). · Zbl 0762.11035
[6] Yokoi, H.: The fundamental unit and class number one problem of real quadratic fields with prime discriminant. Nagoya Math. J., 120 , 51-59 (1990). · Zbl 0715.11057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.