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Field theory and polynomials Definition

Picard–Vessiot extension

Builds on Differential fieldDerivationField extensionLinear differential equation of order nAlgebraically closed fieldSplitting fieldDifferential Galois theory

Definition

Let (K,∂)(K, \partial) be a differential field of characteristic 00, where ∂\partial is a derivation of KK. Let C={c∈K:∂c=0}C = \set{c \in K : \partial c = 0} be its field of constants, and assume that CC is algebraically closed. Let L(y)=y(n)+an−1y(n−1)+⋯+a1y′+a0yL(y) = y^{(n)} + a_{n-1} y^{(n-1)} + \cdots + a_1 y' + a_0 y with a0,…,an−1∈Ka_0, \dots, a_{n-1} \in K, where y(k)y^{(k)} denotes ∂ky\partial^k y. A Picard–Vessiot extension of KK for LL is a differential field extension E⊇KE \supseteq K such that:

  1. There are y1,…,yn∈Ey_1, \dots, y_n \in E with L(yi)=0L(y_i) = 0 for each ii that are linearly independent over the constants of EE.
  2. EE is generated as a field over KK by y1,…,yny_1, \dots, y_n and all their derivatives, that is, E=K(yi(k):1≤i≤n, 0≤k≤n−1)E = K(y_i^{(k)} : 1 \le i \le n, \ 0 \le k \le n - 1).
  3. The field of constants of EE is CC.

Remarks

Under these hypotheses a Picard–Vessiot extension for LL exists, and it is unique up to an isomorphism of differential fields over KK. The group of automorphisms of EE over KK that commute with ∂\partial is the differential Galois group of LL; it is a linear algebraic group over CC.

Example: take K=ℂ(x)K = \C(x) with ∂=d/dx\partial = d/dx, so C=ℂC = \C, and L(y)=y′−yL(y) = y' - y. The field E=ℂ(x,ex)E = \C(x, e^x) is a Picard–Vessiot extension: y1=exy_1 = e^x solves L(y)=0L(y) = 0, it generates EE over KK, and the constants of ℂ(x,ex)\C(x, e^x) are only the complex numbers.

Non-example: for the same KK and LL, the field E′=ℂ(x,ex,ex2)E' = \C(x, e^x, e^{x^2}) contains the solution exe^x and has constant field ℂ\C, but it fails clause 2. It is strictly larger than the field generated by the solutions and their derivatives, because ex2e^{x^2} does not lie in ℂ(x,ex)\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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Further reading

  1. Web https://en.wikipedia.org/wiki/Picard–Vessiot_theory
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MSC12H05, 34M15
DefinesPicard–Vessiot extension
UsesDifferential field, Derivation, Field extension, Linear differential equation of order n, Algebraically closed field, Splitting field, Differential Galois theory

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