---
tag: H5P06V330V
kind: definition
name: Picard–Vessiot extension
msc: [12H05, 34M15]
defines: TB53Y5NMJD
uses: [GA5PTQGC6G, 3J89REV9WQ, 0EQWVHF3W5, 0QQRP17Z7S, 0PF6TM8TWT, 0KXQKX9P63, EEZS4H8QPH]
sources:
  - url: https://en.wikipedia.org/wiki/Picard–Vessiot_theory
---

## Definition

Let $(K, \partial)$ be a [[differential field|GA5PTQGC6G]] of characteristic $0$, where $\partial$ is a [[derivation|3J89REV9WQ]] of $K$.
Let $C = \set{c \in K : \partial c = 0}$ be its field of constants, and assume that $C$ is [[algebraically closed|0PF6TM8TWT]].
Let
$$L(y) = y^{(n)} + a_{n-1} y^{(n-1)} + \cdots + a_1 y' + a_0 y$$
with $a_0, \dots, a_{n-1} \in K$, where $y^{(k)}$ denotes $\partial^k y$.
A *Picard–Vessiot extension* of $K$ for $L$ is a differential [[field extension|0EQWVHF3W5]] $E \supseteq K$ such that:

1. There are $y_1, \dots, y_n \in E$ with $L(y_i) = 0$ for each $i$ that are linearly independent over the constants of $E$.
2. $E$ is generated as a field over $K$ by $y_1, \dots, y_n$ and all their derivatives, that is, $E = K(y_i^{(k)} : 1 \le i \le n, \ 0 \le k \le n - 1)$.
3. The field of constants of $E$ is $C$.

## Remarks

Under these hypotheses a Picard–Vessiot extension for $L$ exists, and it is unique up to an isomorphism of differential fields over $K$.
The group of automorphisms of $E$ over $K$ that commute with $\partial$ is the differential Galois group of $L$; it is a linear algebraic group over $C$.

Example: take $K = \C(x)$ with $\partial = d/dx$, so $C = \C$, and $L(y) = y' - y$.
The field $E = \C(x, e^x)$ is a Picard–Vessiot extension: $y_1 = e^x$ solves $L(y) = 0$, it generates $E$ over $K$, and the constants of $\C(x, e^x)$ are only the complex numbers.

Non-example: for the same $K$ and $L$, the field $E' = \C(x, e^x, e^{x^2})$ contains the solution $e^x$ and has constant field $\C$, but it fails clause 2.
It is strictly larger than the field generated by the solutions and their derivatives, because $e^{x^2}$ does not lie in $\C(x, e^x)$.

The extension is an analogue of the [[splitting field|0KXQKX9P63]] of a polynomial, and it is the basic object of [[differential Galois theory|EEZS4H8QPH]].