Picard–Vessiot extension

Builds on Differential field Derivation Field extension Linear differential equation of order n Algebraically closed field Splitting field Differential Galois theory

Definition

Let $(K, \partial)$ be a differential field of characteristic $0$ , where $\partial$ is a derivation of $K$ . Let $C = \set{c \in K : \partial c = 0}$ be its field of constants, and assume that $C$ is algebraically closed. 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 $E \supseteq K$ such that:

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 $E$ .
$E$ is generated as a field over $K$ by $y_1, \dots, y_n$ and all their derivatives, that is, $E = K(y_i^{(k)} : 1 \le i \le n, \ 0 \le k \le n - 1)$ .
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 of a polynomial, and it is the basic object of differential Galois theory.

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Algebraically closed field concept
Derivation concept
Differential Galois theory concept
Differential field concept
Field extension concept
Linear differential equation of order n concept
Splitting field concept
Picard–Vessiot extension definition

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Tag	`H5P06V330V`
URI	`m2://H5P06V330V`
Kind	definition
Writing	W0 stub
MSC	12H05, 34M15
Defines	Picard–Vessiot extension
Uses	Differential field, Derivation, Field extension, Linear differential equation of order n, Algebraically closed field, Splitting field, Differential Galois theory
Changed	2 November 2026

Picard–Vessiot extension

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