  Let E be a division ring and G a finite group of automorphisms of E whose
elements are distinct modulo inner automorphisms of E. Given a representation
\rho: B-> GL(d,E) of an F-algebra B, we give necessary and sufficient
conditions for \rho to be {\it writable} over F=E^G, i.e. whether or not there
exists a matrix A in GL(d,E) that conjugates \rho(B) into GL(d,F). We give an
algorithm for constructing an A, or proving that no A exists. The case of
particular interest to us is when E is a field, and \rho is absolutely
irreducible. The algorithm relies on an explicit formula for A, and a
generalization of Hilbert's Theorem 90 (Theorem~3) that arises in Galois
cohomology. The algorithm has applications to the construction of absolutely
irreducible group representations (especially for solvable groups), and to the
recognition of one of the classes in Aschbacher's matrix group classification
scheme.
