Cyclicity of the 2-decomposed unramified Iwasawa module

Abstract

Let k be a real quadratic number field, and k∞ its cyclotomic Z2-extension. We study the cyclicity of the Galois group X∞′ over k∞ of the maximal abelian unramified 2-extension, in which all 2-adic primes of k∞ split completely. As consequence, we determine the complete list of real quadratic number fields for which X∞′ is cyclic.When X∞′ is cyclic non-trivial, we give a new infinite family of real quadratic number fields, for which Greenberg's conjecture is valid.