Differential Galois theory

Summary

Differential Galois theory studies linear differential equations over a differential field, see the definition. It uses the group of symmetries of their solutions, as Galois theory does for polynomial equations. The solutions generate a Picard–Vessiot extension, and the differential Galois group is the group of its automorphisms that commute with the derivation. This group is a linear algebraic group over the constants, and there is a Galois correspondence between its closed subgroups and the intermediate differential fields. The theory explains why some equations, and some integrals such as $\int e^{-x^2} \, dx$, cannot be solved in elementary terms, in line with Liouville's theorem.