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Abstract: Let p be an odd prime. Let k be an algebraic number field and let be the compositum of all the -extensions of k, so that for some finite d. We shall consider fields k with . 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. [Copyright &y& Elsevier]