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June 2001 A note on the $\mathbf {Z}_p \times \mathbf {Z}_q$-extension over $\mathbf {Q}$
Kuniaki Horie
Proc. Japan Acad. Ser. A Math. Sci. 77(6): 84-86 (June 2001). DOI: 10.3792/pjaa.77.84


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 $\mathbf{Q}^S$.


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Kuniaki Horie. "A note on the $\mathbf {Z}_p \times \mathbf {Z}_q$-extension over $\mathbf {Q}$." Proc. Japan Acad. Ser. A Math. Sci. 77 (6) 84 - 86, June 2001.


Published: June 2001
First available in Project Euclid: 24 May 2006

zbMATH: 1005.11060
MathSciNet: MR1842861
Digital Object Identifier: 10.3792/pjaa.77.84

Primary: 11R20
Secondary: 11R23

Keywords: Hilbert class field , Iwasawa theory

Rights: Copyright © 2001 The Japan Academy

Vol.77 • No. 6 • June 2001
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