Writing representations over proper division subrings

Let E be a division ring, and G a finite group of automorphisms of E whose elements are distinct modulo inner automorphisms of E. Let F=EG be the division subring of elements of E fixed by G. Given a representation ρ:A→Ed×d of an F-algebra A, we give necessary and sufficient conditions for ρ to be writable over F. (Here Ed×d denotes the algebra of d×d matrices over E, and a matrix A writes ρ over F if A−1ρ(A)A⊆Fd×d.) 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 ρ is absolutely irreducible. The algorithm relies on an explicit formula for A, and a generalization of Hilbert's Theorem 90 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 class C5 in Aschbacher's matrix group classification scheme [M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984) 469–514, MR0746539; Shangzhi Li, On the subgroup structure of classical groups, in: Group Theory in China, in: Math. Appl. (China Ser.), vol. 365, Kluwer Acad. Publ., Dordrecht, 1996, pp. 70–90, MR1447199. [1,13]].