Galois Correspondence Theorem for Picard-Vessiot Extensions

Abstract

For a homogeneous linear differential equation defined over a differential field K , a Picard-Vessiot extension is a differential field extension of K differentially generated by a fundamental system of solutions of the equation and not adding constants. When K has characteristic 0 and the field of constants of K is algebraically closed, it is well known that a Picard-Vessiot extension exists and is unique up to K -differential isomorphism. In this case the differential Galois group is defined as the group of K -differential automorphisms of the Picard-Vessiot extension and a Galois correspondence theorem is settled. Recently, Crespo, Hajto and van der Put have proved the existence and unicity of the Picard-Vessiot extension for formally real (resp. formally p-adic) differential fields with a real closed (resp. p-adically closed) field of constants. This result widens the scope of application of Picard-Vessiot theory beyond the complex field. It is then necessary to give an accessible presentation of Picard-Vessiot theory for arbitrary differential fields of characteristic zero which eases its use in physical or arithmetic problems. In this paper, we give such a presentation avoiding both the notions of differential universal extension and specializations used by Kolchin and the theories of schemes and Hopf algebras used by other authors. More precisely, we give an adequate definition of the differential Galois group as a linear algebraic group and a new proof of the Galois correspondence theorem for a Picard-Vessiot extension of a differential field with non algebraically closed field of constants, which is more elementary than the existing ones.

Some Definable Galois Theory and Examples

“…The Galois correspondence is between algebraic subgroups of G defined over C K and differential fields in between K and L. Theorem 2.3 in this context is contained in [10,Chapter VI] in the slightly more general form of strongly normal extensions. In any case what we have described here subsumes [5,Theorem 4.4] and [4]. The previous paragraph extends to the partial case m > 1, where we now consider a set 1 y = A 1 y, .…”

Real Liouvillian Extensions of Partial Differential Fields

“…K ⊂ E ⊂ L, then L|E is a real Picard-Vessiot extension and DGal(L|E) is a C K -defined closed subgroup of DGal(L|K). As in the ordinary case (see [3] Theorem 1), we obtain a Galois correspondence theorem.…”

Real Liouvillian Extensions of Partial Differential Fields

“…As in the ordinary case (see [8]), we shall consider the set DHom K L, L of K-differential morphisms from L into L and transfer the group structure from DAut K L to DHom K L, L by means of the bijection…”