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Real and $$p$$-adic Picard-Vessiot fields. (English) Zbl 1344.34096
The authors consider differential modules $$M$$ over real and $$p$$-adic differential fields $$K$$ (with fields of constants $$k$$ real closed or $$p$$-adic closed). A Picard-Vessiot field (PVF) $$L$$ for $$M/K$$ is a field extension of $$K$$ with the same field of constants $$k$$, where $$L$$ is equipped with a differentiation extending the one of $$K$$ and there exists an invertible $$d\times d$$-matrix $$F$$ ($$d=$$dim$$M$$) with entries in $$L$$ satisfying $$F'=AF$$, as a field $$L$$ being generated over $$K$$ by the entries of $$F$$. Using results of J.-P. Serre and P. Deligne, the authors obtain a purely algebraic proof of the existence and unicity of PVFs. They treat the inverse problem for real forms of a semisimple group and they give examples illustrating the relations between differential modules, PVFs and real forms of a linear algebraic group.

##### MSC:
 12H20 Abstract differential equations 12H25 $$p$$-adic differential equations 34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain 11E10 Forms over real fields 11R34 Galois cohomology
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