Solvable-by-finite groups as differential Galois groups
Résumé
We prove the inverse problem of differential Galois theory over the differential field k=C(x), where C is an algebraic closed field of characteristic zero, for linear algebraic groups G over CC with a solvable identity component G°. We show that for any k-irreducible principal homogeneous space V for G, the derivation d/dx of k can be extended on k(V) in such a way that k(V) is a Picard-Vessiot extension of k with Galois group G. The proof is constructive up to the finite embedding problem of classicalGalois theory over C(x).
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