Picard-Vessiot Extension over a Differential Field?

[KarmonEuloid]

Given a differential field F and a linear algebraic group G over the constant field C of F, find a Picard-Vessiot extension of E of F with G(E/F)=G:

This isn't homework, just something I saw in a book that I was curious about. The author says that this can be shown but doesn't illustrate how. Can anyone help?

 

[morphism]

Caveat: I know nothing about this subject. I checked Wikipedia for the relevant definitions, and I believe this works. First find a Picard-Vessiot extension E/F with G(E/F)=GL_n(C) (such a thing does exist, right??). Next, given a linear algebraic group G, view it as sitting in some GL_n(C)=G(E/F), and then consider the fixed field E^G (this notion makes sense, right??). If the Galois theory of Picard-Vessiot extensions works like normal Galois theory (i.e. if you have an analogue of Artin's theorem), then E/E^G should be Picard-Vessiot and G(E/E^G) should be G.

Note that this proof is identical to the standard proof that every finite group G is the Galois group of some extension. (The role of GL_n above is played by S_n here.)

 

[KarmonEuloid]

Caveat: I know nothing about this subject. I checked Wikipedia for the relevant definitions, and I believe this works. First find a Picard-Vessiot extension E/F with G(E/F)=GL_n(C) (such a thing does exist, right??). Next, given a linear algebraic group G, view it as sitting in some GL_n(C)=G(E/F), and then consider the fixed field E^G (this notion makes sense, right??). If the Galois theory of Picard-Vessiot extensions works like normal Galois theory (i.e. if you have an analogue of Artin's theorem), then E/E^G should be Picard-Vessiot and G(E/E^G) should be G.

Note that this proof is identical to the standard proof that every finite group G is the Galois group of some extension. (The role of GL_n above is played by S_n here.)

This seems to work to me. I reposted it on MathOverflow (giving you credit of course) to see if they could verify it (as that is where the question originally came from), although I would also appreciate it if someone here could check this or provide an alternate answer. Thanks.

 

[morphism]

I read a bit more about this topic and can now confirm that the above proof is correct, provided the field of constants C is algebraically closed. This assumption is apparently necessary for the analogue of Artin's theorem to hold for Picard-Vessiot extensions. See Chapter 6 of Crespo and Hajto, Algebraic Groups and Differential Galois Theory (AMS 2011), freely available here. [Also note: Exercise 7 shows that, for any n, there is a Picard-Vessiot extension E/F with G(E/F)=GL_n(C).]

I don't know what happens if C is not algebraically closed.

 

[blue_raver22]

The Picard-Vessiot extension over a differential field is a fundamental concept in differential Galois theory. It is a field extension that allows us to study the solutions of a differential equation using the tools of algebraic geometry and group theory.

In the given scenario, we have a differential field F and a linear algebraic group G over the constant field C of F. We want to find a Picard-Vessiot extension E of F with G(E/F)=G. This essentially means that the group of automorphisms of E over F is isomorphic to the linear algebraic group G.

To find such an extension, we need to consider the differential equations that are defined over F and have G as their differential Galois group. These are called Picard-Vessiot differential equations.

The key idea is to construct a differential field extension of F that is generated by solutions of the Picard-Vessiot differential equation. This extension, denoted by E, will be the desired Picard-Vessiot extension.

To show that G(E/F)=G, we need to prove that any automorphism of E over F can be extended to an automorphism of the Picard-Vessiot differential equation. This can be done by using the Galois theory of differential fields.

In summary, the Picard-Vessiot extension over a differential field allows us to study differential equations using the powerful tools of algebraic geometry and group theory. It is a crucial concept in differential Galois theory and has many applications in the study of differential equations.