Proceedings of the Japan Academy, Series A, Mathematical Sciences

A note on the $\mathbf {Z}_p \times \mathbf {Z}_q$-extension over $\mathbf {Q}$

Kuniaki Horie

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Let $S$ be a non-empty set of prime numbers; $1 \leq |S| \leq \infty$. Let $\mathbf{Q}^S$ denote the abelian extension of the rational field $\mathbf{Q}$ whose Galois group over $\mathbf{Q}$ is topologically isomorphic to the direct product of the additive groups of $l$-adic integers for all $l \in S$. In this note, we shall give simple examples of $S$ such that, for some $l \in S$, the Hilbert $l$-class field over $\mathbf{Q}^S$ is a nontrivial extension of $\mathbf{Q}^S$. Our results imply that, if $S$ contains 2, 3, 31, and 73, then there exists an unramified cyclic extension of degree $2263 = 31 \cdot 73$ over $\mathbf{Q}^S$.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 77, Number 6 (2001), 84-86.

First available in Project Euclid: 24 May 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R20: Other abelian and metabelian extensions
Secondary: 11R23: Iwasawa theory

Hilbert class field Iwasawa theory


Horie, Kuniaki. A note on the $\mathbf {Z}_p \times \mathbf {Z}_q$-extension over $\mathbf {Q}$. Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), no. 6, 84--86. doi:10.3792/pjaa.77.84.

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