Abstract
We consider differential modules over real and p-adic differential fields K such that its field of constants k is real closed (resp., p-adically 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.
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