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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 {\it 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 ${\bbfQ}(\sqrt{n\sp 2-a\sp t})$ by t, which are a generalization of works of {\it M. J. Cowles} [J. Number Theory 12, 113-115 (1980; Zbl 0427.12001)] and {\it B. H. Gross} and {\it D. E. Rohrlich} [Invent. Math. 44, 201-224 (1978; Zbl 0369.14011)].
Reviewer: H.Yokoi

MSC:
11R11Quadratic extensions
11R23Iwasawa theory
11D09Quadratic and bilinear diophantine equations
WorldCat.org
Full Text: DOI
References:
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[12] Takeuchi, H.: On the class-number of the maximal real subfield of a cyclotomic field. Canad. J. Math. 33, No. No. 1, 55-58 (1981) · Zbl 0482.12004
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[14] Yokoi, H.: On the Diophantine equation $x2-py2={\pm}$4q and the class number of real subfields of a cyclotomic field. Nagoya math. J. 91, 151-161 (1983) · Zbl 0506.10012