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In this paper we present a reformulation of the Galois correspondence theorem of Hopf Galois theory in terms of groups carrying farther the description of Greither and Pareigis. We prove that the class of Hopf Galois extensions for which the Galois correspondence is bijective is larger than the class of almost classically Galois extensions but not equal to the whole class. We show as well that the image of the Galois correspondence does not determine the Hopf Galois structure.

We consider differential modules over real and p-adic differential fields K such that its field of constants k is real closed ( resp., padically closed). Using P. Deligne's work on Tannakian categories and a result of J.-P. Serre on Galois cohomology, a purely algebraic proof of the existence and unicity of real (resp., p-adic) Picard-Vessiot fields is obtained. The inverse problem for real forms of a semi-simple group is treated. Some examples illustrate the relations between differential modules, Picard-Vessiot fields and real forms of a linear algebraic group. 1