Central simple p-algebras with purely inseparable subfields

DESCENT THEOREM. Let C be a simple algebra with center E, and let G be a $nite group of automorphism of E with fixed field F. If F can be faithfully lifted to a group G of E-semilinear automorphisms of C (i.e. for CT E G, L? is an automorphism of C such that O(xv) = a(x) Z(v) for x E E and v E C), then C g C’ OF E where C’ is the algebra of fixed elements of G. Moreover, C’ is simple with center F, and if [C : E] < 00, then [C : E] = [C’ : F].