Greenbergʹs conjecture and capitulation in image-extensions Original Research Article

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