Stabilization on ideal class groups in potential cyclic towers

Let p be a prime and let F be a number field. Consider a Galois extension $K/F$ with Galois group H Δ where H Zₚ or Z/pᵈZ, and Δ is an arbitrary Galois group. The subfields fixed by Hpⁿ Δ (n=0,1,⋯) form a tower which we call it a potential cyclic p-tower in this paper. A radical p-tower is a typical example, say Z⊂ Z(√[p]a)⊂ Z(√[p²]a)⊂ ⋯ where a∈ Z. We extend the stabilization result of Fukuda in Iwasawa theory on p-class groups in cyclic p-towers to potential cyclic p-towers. We also extend Iwasawa's class number formula in Zₚ-extensions to potential Zₚ-extensions.