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On Ono’s problem for quadratic fields. (English) Zbl 0780.11050
Für einen quadratischen Zahlkörper \(k\) sei \(d_ k\) die Diskriminante, \(h_ k\) die Klassenzahl und \(M_ k\) die Minkowski- Schranke von \(k\) (\(M_ k=\sqrt{d_ k}/2\), falls \(k\) reell; \(M_ k=2\sqrt{-d_ k}/\pi\), falls \(k\) imaginär). Der Autor bestimmt (unter Voraussetzung von GRH) alle quadratischen Zahlkörper \(k\) mit folgender Eigenschaft: \(h_ k\) ist ungerade, und \((d_ k/p)\neq 1\) für alle Primzahlen \(p\leq M_ k\) (es gibt 42 solche Körper). Resultate von ähnlichem Typ wurden von S. Louboutin, R. A. Mollin und H. C. Williams erzielt [siehe Can. J. Math. 44, 824-842 (1992; Zbl 0771.11039)] und die dort zitierte Literatur.
11R11 Quadratic extensions
11R29 Class numbers, class groups, discriminants
Full Text: DOI
[1] M. G. Leu : On a conjecture of Ono on real quadratic fields. Proc. Japan Acad., 63A, 323-326(1987). · Zbl 0659.12004
[2] M. G. Leu : On a problem of Ono and quadratic non-residue. Nagoya Math. J., 115, 185-198 (1989). · Zbl 0659.12005
[3] R. A. Mollin and H. C. Williams: Quadratic non-residue and prime-producing polynomials. Canada. Math. Bull., 32, 474-478 (1989). · Zbl 0714.11066
[4] Oesterle: Versions effectives du theorem de Chebotalev sous L’Hypothese de Riemann Generalise. Soc. Math. France Ast6risque, 61, 165-167 (1979). · Zbl 0418.12005
[5] T. Ono: A problem on quadratic fields. Proc. Japan Acad., 64A, 78-79 (1988). · Zbl 0662.12004
[6] J. B. Rosser and L. Schoenfeld: Approxiamate formulas for some functions of prime numbers. Illinois J. Math., 6, 64-94 (1962). · Zbl 0122.05001
[7] H. M. Stark: A complete determination of the complex quadratic fields of class number one. Michigan Math. J., 14, 1-27 (1967). · Zbl 0148.27802
[8] K. Yosidome and Y. Asaeda: On Ono’s problem on quadratic fields. Proc. Japan Acad., 67A, 348-352 (1991). · Zbl 0746.11046
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