In this paper we study the question of when an $H$-comodule algebra is a faithfully flat Galois extension of its subalgebra of coinvariants for certain Hopf algebras $H$. We note that if $H$ is connected, a faithfully flat Galois extension must actually be cleft and hence a crossed product, and we show that with a different hypothesis, a faithfully flat Galois extension must be a smash product. We also describe faithfully flat Galois extensions when $H$ is pointed cococummutative. We give an explicit description of $H$-comodule algebras when $H$ is a polynomial algebra, a divided power Hopf algebra, a free algebra, or a shuffle algebra. We give necessary and sufficient conditions for an $H$-extension to be faithfully flat Galois in these cases and in the case where $H$ is the enveloping algebra of a Lie algebra; a key ingredient in our analysis is the existence and description of a total integral. In the case where $H=k[x]$, we give a simple example of a flat Galois extension that is not faithfully flat.
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