---
tag: TB53Y5NMJD
kind: concept
name: Picard–Vessiot extension
msc: [12H05, 34M15]
specializes: [0EQWVHF3W5]
sources:
  - url: https://en.wikipedia.org/wiki/Picard–Vessiot_theory
---

## Summary

A Picard–Vessiot extension is the smallest [[differential field|GA5PTQGC6G]] extension that contains a full set of solutions of a linear differential equation.
It adds no new constants.
It plays the role for linear differential equations that the splitting field plays for polynomial equations.
Its group of differential automorphisms is the differential Galois group of the equation, the main object of [[differential Galois theory|EEZS4H8QPH]].

## Explanations

### L1 human

Start with a field of functions, such as the rational functions $\C(x)$, and a [[linear differential equation|0QQRP17Z7S]] with coefficients in it.
Often the solutions are not in the field, so one adjoins them together with all their derivatives.
The result is a bigger field in which the equation can be solved completely.
For $y' = y$ over $\C(x)$, one adjoins $e^x$ and gets the field $\C(x, e^x)$.
The symmetries of this bigger field that fix $\C(x)$ and commute with differentiation tell how the solutions are related to each other.