Class number behavior in a two-tiered tower of ℤp-extensions

Let [Formula: see text] be the cyclotomic [Formula: see text]-extension field of an algebraic number field k. Moreover, we take a [Formula: see text]-extension [Formula: see text] over [Formula: see text]. In this paper, we study the behavior of the p-part of the class number of intermediate fields of [Formula: see text], and the structure of the unramified Iwasawa module [Formula: see text] as a module over [Formula: see text]. We show that a formula on the p-part of the class number of certain intermediate fields of [Formula: see text] holds under several assumptions (one of these assumptions is that [Formula: see text] is a finitely generated torsion [Formula: see text]-module). We also consider the structure of [Formula: see text] for the case that every prime lying above p is unramified in [Formula: see text]. In particular, we give sufficient conditions and examples such that the above assumptions are satisfied.