On the ℓ-part of the Zp1×⋯×Zps-extension of Q

Let S be a non-empty finite set of prime numbers and QS the abelian number field whose Galois group is topologically isomorphic to the direct product of the p-adic integer rings for all p in S. We denote by ΩS the composite of pn-th cyclotomic fields for all p in S and all positive integers n. Let ℓ be a prime number which is not in S and we define an explicit constant which depends only on S and the decomposition field F of ℓ for ΩS over Q. Then ℓ does not divide the class numbers of all intermediate fields of QS with finite degree over Q if ℓ is greater than . Our proof refines Horieʼs method.