## 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

Zbl 0222.12003
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