  We show that a finite-dimensional tame division algebra D over a Henselian
field F has a maximal subfield Galois over F if and only if its residue
division algebra has a maximal subfield Galois over the residue field of F.
  This generalizes the mechanism behind several known noncrossed product
constructions to a crossed product criterion for all tame division algebras,
and in particular for all division algebras if the residue characteristic is 0.
If the residue field is a global field, the criterion leads to a description of
the location of noncrossed products among tame division algebras, and their
discovery in new parts of the Brauer group.
