PicardVessiot

Picard-Vessiot theory is the differential Galois theory for linear differential equations. It studies the algebraic symmetries of the solution space by attaching a differential field extension that encodes all solutions. The theory, named after Charles Émile Picard and Ernest Vessiot, provides a Galois-theoretic framework for questions of solvability and expression by quadratures.

Setup and definition: Let F be a differential field of characteristic zero with derivation D, and let C be its field of constants, C = {c in F : D(c) = 0}. For a system of linear differential equations y' = A y with A ∈ M_n(F), a Picard-Vessiot extension K/F is a differential field extension such that (i) K is generated over F by a fundamental solution matrix Y with Y' = A Y and det Y ≠ 0, (ii) the field of constants of K is exactly C, and (iii) K contains no new constants beyond those in F. If the constant field C is algebraically closed, such an extension exists and is unique up to F-differential isomorphism.

Differential Galois group: The differential Galois group G = AutΔ(K/F) consists of all F-automorphisms of K that commute with the derivation. Each σ ∈ G is determined by σ(Y) = Y cσ for some cσ ∈ GL_n(C), so G embeds as a linear algebraic subgroup of GL_n(C) defined over C. There is a Galois correspondence between intermediate differential fields F ⊆ L ⊆ K and algebraic subgroups H ⊆ G.

Significance: Picard-Vessiot theory generalizes classical Galois theory to differential equations and provides criteria for solvability in terms of the structure of the Galois group. In particular, the equation is solvable in terms of Liouvillian functions if and only if its PV group is a solvable algebraic group. The theory, developed in the 1890s, remains central in differential algebra and the study of differential equations.