Let A be the inverse limit of the p-part of the ideal class groups in a ℤrp-extension K∞/K. Greenberg conjectures that if r is maximal, then A is pseudo-null as a module over the Iwasawa algebra A (that is, has codimension at least 2). We prove this conjecture in the case that K is the field of pth roots of unity, p has index of irregularity 1, satisfies Vandiver's conjecture, and satisfies a mild additional hypothesis on units. We also show that if K is the field of pth roots of unity and r is maximal, Greenberg's conjecture for K implies that the maximal p-ramified pro-p-extension of K cannot have a free pro-p quotient of rank r unless p is regular. Finally, we prove a generalization of a theorem of Iwasawa in the case r = 1 concerning the Kummer extension of K∞ generated by p-power roots of p-units. We show that the Galois group of this extension is torsion-free as a Λ-module if there is only one prime of K above p and K∞ contains all the p-power roots of unity.
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