Galois subfields of inertially split division algebras

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 (D) over bar contains a maximal subfield which is Galois over the residue field (F) over bar. 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. (C) 2011 Elsevier Inc. All rights reserved.