We bring together ideas in analysis of Hopf *-algebra actions on II_1
subfactors of finite Jones index and algebraic characterizations of Frobenius,
Galois and cleft Hopf extensions to prove a non-commutative algebraic analogue
of the classical theorem: a finite field extension is Galois iff it is
separable and normal. Suppose N < M is a separable Frobenius extension of
k-algebras split as N-bimodules with a trivial centralizer C_M(N). Let M_1 :=
End(M)_N and M_2 := End(M_1)_M be the endomorphism algebras in the Jones tower
N < M < M_1 < M_2. We show that under depth 2 conditions on the second
centralizers A := C_{M_1}(N) and B : = C_{M_2}(M) the algebras A and B are
semisimple Hopf algebras dual to one another and such that M_1 is a smash
product of M and A, and that M is a B-Galois extension of N.
