  We describe Greenberg's pseudo-null conjecture, and prove a result describing
conditions under which the pseudo-null conjecture for a number field $K$
implies the conjecture for finite extensions of $K$. We then apply the result
to the cyclotomic $\mathbb{Z}_p$-tower above a cyclotomic field of prime roots
of unity, verifying the conjecture for a large class of cyclotomic fields.
