  Let F denote an unramified extension of the cyclotomic extension of Q_p by
(p^n)th roots of unity, for an odd prime p. We determine the conductors of
those Kummer extensions of F of degree dividing p^n which are Galois over the
maximal unramified subfield of F. This follows from a comparison of the Galois
module structure of the multiplicative group of F with the unit filtration of
F.
