Abelian 2-class field towers over the cyclotomic $${\mathbb {Z}_2}$$ -extensions of imaginary quadratic fields

Abstract

For the cyclotomic \({\mathbb Z_2}\)-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 \({\mathbb {Z}_2}\)-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.