Cyclicity of the 2-decomposed unramified Iwasawa module

Let k be a real quadratic number field, and $k_{\infty }$ its cyclotomic $Z_2$-extension. We study the cyclicity of the Galois group $X_{\infty }^{\prime }$ over $k_{\infty }$ of the maximal abelian unramified 2-extension, in which all 2-adic primes of $k_{\infty }$ split completely. As consequence, we determine the complete list of real quadratic number fields for which $X_{\infty }^{\prime }$ is cyclic.

When $X_{\infty }^{\prime }$ is cyclic non-trivial, we give a new infinite family of real quadratic number fields, for which Greenberg's conjecture is valid.