---
tag: EEZS4H8QPH
kind: concept
name: Differential Galois theory
msc: [12H05, 34M15]
sources:
  - url: https://en.wikipedia.org/wiki/Differential_Galois_theory
same_as:
  wikidata: Q2383624
  wikipedia: Differential Galois theory
---

## Summary

Differential Galois theory studies linear differential equations over a [[differential field|GA5PTQGC6G]], see the [[definition|RW8DQQYJMR]].
It uses the group of symmetries of their solutions, as [[Galois theory|WNQDKQYZ2R]] does for polynomial equations.
The solutions generate a [[Picard–Vessiot extension|TB53Y5NMJD]], 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|8VRTPECBPZ]].

## Explanations

### L1 human

Classical Galois theory explains why the quintic equation has no formula in radicals, by looking at the symmetries of its roots.
Differential Galois theory does the same for linear differential equations such as $y'' + x y = 0$.
The symmetries now form a group of matrices that can be continuous, such as all rotations or all scalings.
For $y' = y$ over the rational functions, the solutions are $c e^x$, and the symmetries are the scalings $e^x \mapsto c e^x$ with $c \ne 0$.
Whether an equation can be solved by integrals, exponentials and algebraic functions depends on whether this group is solvable in a suitable sense.