Abelian 2-class field towers over the cyclotomic ℤ₂-extensions of imaginary quadratic fields

For the cyclotomic ℤ₂-extension k_∞ of an imaginary quadratic field k, we consider whether the Galois group G(k_∞) of the maximal unramified pro-2-extension over k_∞ is abelian or not. The group G(k_∞) is abelian if and only if the nth layer of the ℤ₂-extension has abelian 2-class field tower for all n ≥ 1. The purpose of this paper is to classify all such imaginary quadratic fields k in part by using Iwasawa polynomials.