On the \(\ell \)-part of the \(\mathbb Z_{p_1}\times \dots \times \mathbb Z_{p_s}\)-extension of \(\mathbb Q\)
Let \(S\subset \mathbb{N}\) be a non-empty finite set of prime numbers. Let \(\mathbb{Q}^S/\mathbb{Q}\) be the abelian extension whose Galois group is topologically isomorphic to the direct product \(\prod_{p\in S}\mathbb{Z}_p\), where \(\mathbb{Z}_p\) is the ring of \(p\)-adic integers. We denote by \(\Omega_S\) the compositum of the \(p^n\)-th cyclotomic fields over \(\mathbb{Q}\), where \(p\) runs through \(S\) and \(n\) through \( \mathbb{N}\). Let \(l\) be a prime number such that \(l\not \in S\). Let \(F\) be the decomposition field of \(l\) in the extension \(\Omega_S/\mathbb{Q}\). The author defines an explicit constant \(C\) which depends only on S and \(F\). He proves that if \(l>C\), then \(l\) does not divide the class numbers of all intermediate fields of \(\mathbb{Q}^S/\mathbb{Q}\) with finite degree over \(\mathbb{Q}\) . The main result improves Theorem 1 in [\textit{K. Horie}, J. Math. Soc. Japan 59, No. 3, 811--824 (2007; Zbl 1128.11052)] by giving a better constant. The proof which refines Horie's method uses among other things: circular units, height of algebraic numbers and Minkowski's convex body theorem.
scientific article; zbMATH DE number 6154093
scientific article