  We study Galois extensions Coinv(M)<M for M an H-comodule algebra and H a
Frobenius Hopf algebroid. We obtain generalizations of various theorems in
Hopf-Galois theory by Kreimer-Takeuchi, Doi-Takeuchi and
Cohen-Fischman-Montgomery. An algebra extension is Galois precisely if it is
balanced, depth 2, and Frobenius. Then we show that Yetter-Drinfeld categories
over H are always braided and their braided commutative algebras play the role
of noncommutative scalar extensions by the Brzezinski-Militaru Theorem.
Contravariant "fiber functors" are used to prove an analogue of Ulbrich's
Theorem and to get a monoidal embedding of the category of modules over the
endomorphism Hopf algebroid E=End(_N M_N).
