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Diophantine equations and class numbers. (English) Zbl 0591.12006

In this paper, the author intends to extend some already known results. First he tries to provide sufficient conditions for non-trivial class numbers of real quadratic fields of Richaud-Degert type, in order to generalize results of Ankeny-Chowla-Hasse, S. D. Lang, H. Takeuchi and I. Yamaguchi. However, almost all of them essentially have been already obtained by H. Hasse [Elem. Math. 20, 49-59 (1965; Zbl 0128.035)] and the reviewer [Nagoya Math. J. 91, 151-161 (1983; Zbl 0506.10012)].

Next he obtains necessary and sufficient conditions for an algebraic integer, not a unit, to be the norm of an algebraic integer in a given extension field. By using this, finally, he gives sufficient conditions for the divisibility of the class numbers of imaginary quadratic fields $ℚ\left(\sqrt{{n}^{2}-{a}^{t}}\right)$ by t, which are a generalization of works of M. J. Cowles [J. Number Theory 12, 113-115 (1980; Zbl 0427.12001)] and B. H. Gross and D. E. Rohrlich [Invent. Math. 44, 201-224 (1978; Zbl 0369.14011)].

Reviewer: H.Yokoi

##### MSC:
 11R11 Quadratic extensions 11R23 Iwasawa theory 11D09 Quadratic and bilinear diophantine equations