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

Abstract

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 $\mathbf{Q}^S$.