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A note on the divisibility of class numbers of real quadratic fields. (English) Zbl 1036.11057

Y. Yamamoto [Osaka J. Math. 7, 57–76 (1970; Zbl 0222.12003)] and P. J. Weinberger [J. Number Theory 5, 237–241 (1973; Zbl 0287.12007)] independently proved by different constructive methods that for any natural number \(g\geqq 3\) there exist infinitely many real quadratic fields whose class groups contain an element of order \(g\) [cf. H. Cohen and H. W. Lenstra jun., Lect. Notes Math. 1068, 33–62 (1984; Zbl 0558.12002)]. It is expected that for the number \(S_{g}(X)\) of such fields with discriminant \(\leqq X\) we have \(S_{g}(X)\sim C_{g} X\) for some positive constant \(C_{g}\), however, the asymptotic formula for \(S_{g}(X)\) is still unknown even for a single \(g\geqq 3\). M. Ram Murty [Topics in number theory, Kluwer Acad. Publishers. Math. Appl., Dordr. 467, 229–239 (1999; Zbl 0993.11059)] made a result of Weinberger effective to obtain \(S_{g}(X)\gg X^{1/2g-\varepsilon}\) for odd \(g\). The author improves this to \(S_{g}(X)\gg X^{1/g-\varepsilon}\) by estimating the number of discriminants based on a work of Y. Yamamoto. Similar results are known also for imaginary quadratic fields.

MSC:

11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
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[1] Ankeny, N.; Chowla, S., On the divisibility of the class numbers of quadratic fields, Pacific J. Math., 5, 321-324 (1955) · Zbl 0065.02402
[2] H. Cohen, and, H. W. Lenstra, Heuristics on Class Groups of Number Fields, Springer Lecture Notes, in; H. Cohen, and, H. W. Lenstra, Heuristics on Class Groups of Number Fields, Springer Lecture Notes, in
[3] Davenport, H.; Heilbronn, H., On the density of discriminants of cubic fields II, Proc. Roy. Soc. A, 322, 405-420 (1971) · Zbl 0212.08101
[4] Greaves, G., Power-free values of binary forms, Quart. J. Math. Oxford, 43, 45-65 (1992) · Zbl 0768.11034
[5] M. R. Murty, The ABCin; M. R. Murty, The ABCin · Zbl 0893.11043
[6] Murty, M. R., Exponents of class groups of quadratic fields, Topics in Number Theory, (University Park, PA, 1997), 467 (1999), Kluwer Acad. Publ: Kluwer Acad. Publ Dordrecht, p. 229-239 · Zbl 0993.11059
[7] Nagell, T., Über die Klassenzahl imaginar quadratischer Zahlkörper, Abh. Math. Seminar Univ. Hamburg, 1, 140-150 (1922) · JFM 48.0170.03
[8] Ono, K., Indivisibility of class numbers of real quadratic fields, Compositio Math., 119, 1-11 (1999) · Zbl 1002.11080
[9] Soundararajan, K., Divisibility of class numbers of imaginary quadratic fields, J. London Math. Soc., 61, 681-690 (2000) · Zbl 1018.11054
[10] Weinberger, P., Real quadratic fields with class numbers divisible by \(n\), J. Number Theory, 5, 237-241 (1973) · Zbl 0287.12007
[11] Yamamoto, Y., On unramified Galois extensions of quadratic number fields, Osaka J. Math., 7, 57-76 (1970) · Zbl 0222.12003
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