  In this paper we prove Greenberg's pseudo-null conjecture for the field of
p-th roots of unity in the case that p exactly divides the class number and the
index of the global units in the local units. We also generalize to the case of
multiple Z_p-extensions a theorem of Iwasawa on the Kummer extension of the
cyclotomic Z_p-extension generated by p-power roots of units .
