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Parametrization of the quadratic fields whose class numbers are divisible by three. (English) Zbl 0976.11050

The authors show that every quadratic number field \(K\) with class number divisible by \(3\) has the form \(K = \mathbb Q(\sqrt{4uw^3 - 27u^2})\), where \(u, w\) are integers satisfying certain congruence conditions modulo \(27\). This result contains many previous constructions e.g. by P. Hartung [J. Number Theory 6, 279-281 (1974; Zbl 0317.12002)], K. Ohta [Mem. Gifu Tech. Coll. 17, 51-54 (1981)] and J. Brinkhuis [Acta Arith. 69, 1-9 (1995; Zbl 0838.11073)] as special cases.

MSC:

11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
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References:

[1] Brinkhuis, J., Normal integral bases and the spiegelungssatz of Scholz, Acta Arithmetica, 69, 1-9 (1995) · Zbl 0838.11073
[2] Hartung, P., Explicit construction of a class of infinitely many imaginary quadratic fields whose class number is divisible by 3, J. Number Theory, 6, 279-281 (1974) · Zbl 0317.12002
[3] Hendy, M. D., Class number divisors for some real quadratic fields, Occ. Pub. Math., 5, 1-3 (1977) · Zbl 0392.12004
[4] Honda, T., On real quadratic fields whose class numbers are multiples of 3, Reine Angew. Math., 273, 101-102 (1968) · Zbl 0165.06501
[5] Llorente, P.; Nart, E., Effective determination of the decomposition of the rational prime in a cubic field, Proc. American Math. Soc., 87, 579-585 (1983) · Zbl 0514.12003
[6] Mollin, R. A., Solutions of diophantine equations and divisibility of class numbers of complex quadratic fields, Glasgow Math. J., 38, 195-197 (1996) · Zbl 0859.11058
[7] Nakahara, T., On real quadratic fields whose ideal class groups have a cyclic \(p\)-subgroup, Reports Fac. Sci. Eng. Saga Univ. Math., 6, 15-26 (1978) · Zbl 0402.12002
[8] Ohta, K., On algebraic number fields whose class numbers are multiples of 3, Mem. Gifu Tech. Coll., 17, 51-54 (1981)
[9] Uehara, T., On class numbers of imaginary quadratic and quartic fields, Archiv der Math., 41, 256-260 (1983) · Zbl 0532.12007
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