  Let D be a valued division algebra, finite-dimensional over its center F.
Assume D has an unramified splitting field. The paper shows that if D contains
a maximal subfield which is Galois over F (i.e. D is a crossed product) then
the residue division algebra of D contains a maximal subfield which is Galois
over the residue field of F. This theorem captures an essential argument of
previously known noncrossed product proofs in the more general language of
noncommutative valuations. The result is particularly useful in connection with
explicit constructions.
