Greenberg's conjecture and capitulation in -extensions

Let p be an odd prime. Let k be an algebraic number field and let be the compositum of all the Zp-extensions of k, so that for some finite d. We shall consider fields k with Gal(k/Q)≃n(Z/2Z). Building on known results for quadratic fields, we shall show that the Galois group of the maximal abelian unramified pro-p-extension of is pseudo-null for several such k's, thus confirming a conjecture of Greenberg. Moreover we shall see that pseudo-nullity can be achieved quite early, namely in a -extension, and explain the consequences of this on the capitulation of ideals in such extensions.