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On a criterion for the class number of a quadratic number field to be one

Published online by Cambridge University Press:  22 January 2016

Masakazu Kutsuna
Masakazu Kutsuna*
Affiliation:
Department of Liberal Arts, Gifu Technical College
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G. Rabinowitsch [3] generalized the concept of the Euclidean algorithm and proved a theorem on a criterion in order that the class number of an imaginary quadratic number field is equal to one:

Theorem. It is necessary and sufficient for the class number of an imaginary quadratic number fieldD = 1 — 4m, m > 0, to be one that x2 — x + m is prime for any integer x such that 1 ≤ x ≤ m — 2.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 79 , October 1980 , pp. 123 - 129
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1980

References

[ 1 ] Nagel, T., Über die Klassenzahl imaginär-quadratischer Zahlkörper. Abh. Math. Sem. U. Hamburg 1 (1922), 140–150.CrossRefGoogle Scholar
[ 2 ] Perron, O., Die Lehre von den Kettenbrüchen, 2. Auf. Chelsea.Google Scholar
[ 3 ] Rabinowitsch, G., Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern. J. reine angew. Math. 142 (1913), 153–164.CrossRefGoogle Scholar