  A splitting field of a central simple algebra is said to be absolute Galois
if it is Galois over some fixed subfield of the centre of the algebra. The
paper provides an existence theorem for such fields over global fields with
enough roots of unity. As an application, all twisted function fields and all
twisted Laurent series rings over symbol algebras (or p-algebras) over global
fields are crossed products. A closely related statement holds for division
algebras over Henselian valued fields with global residue field.
  The existence of absolute Galois splitting fields in central simple algebras
over global fields is equivalent to a suitable generalization of the weak
Grunwald-Wang Theorem, which is proved to hold if enough roots of unity are
present. In general, it does not hold and counter examples have been used in
noncrossed product constructions. This paper shows in particular that a certain
computational difficulty involved in the construction of explicit examples of
noncrossed product twisted Laurent series rings can not be avoided by starting
the construction with a symbol algebra.
