Abelian 2-class field towers over the cyclotomic Z(2)-extensions of imaginary quadratic fields

摘要

For the cyclotomic Z(2)-extension k(infinity) of an imaginary quadratic field k, we consider whether the Galois group G(k(infinity)) of the maximal unramified pro-2-extension over k(infinity) is abelian or not. The group G(k(infinity)) is abelian if and only if the nth layer of the 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.