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Class numbers of cubic cyclic fields. (English) Zbl 0281.12007

Let \(n\) be any given positive integer. It is known that there exist real (imaginary) quadratic fields whose class numbers are divisible by \(n\). This is classical for the imaginary case and the real case was proved by Y. Yamamoto [ Osaka J. Math. 7, 57–76 (1970; Zbl 0222.12003)]. It is shown in this paper that the same is true for cubic cyclic fields. We deal with fields defined by equations \(X^3+pX^2+2pX+p=0\) for integers \(p\) and special properties of these equations play essential roles.

MSC:

11R29 Class numbers, class groups, discriminants
11R16 Cubic and quartic extensions
11R20 Other abelian and metabelian extensions

Citations:

Zbl 0222.12003
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