Solvable-by-finite groups as differential Galois groups

Claude Mitschi (1) , Michael F. Singer

1 IRMA - Institut de Recherche Mathématique Avancée

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).

Keywords: Algebraic group; Lie algebra; "Algebraic group; principal homogeneous space; differential Galois group; inverse problem; Lie algebra."

Domains: Commutative Algebra [math.AC]

Claude Mitschi, Michael F. Singer. Solvable-by-finite groups as differential Galois groups. 2002. ⟨hal-00129674⟩