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:
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$\begingroup$ This is phrased as homework. Care to try again? $\endgroup$– Will JagyCommented Mar 29, 2012 at 21:32
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$\begingroup$ Some possibly helpful sources: Section 3.4 and Chapter 4 of library.msri.org/books/Book41/files/matzat.pdf, or www4.ncsu.edu/~singer/papers/COOK_MITSCHI_SINGER.PDF $\endgroup$– B RCommented Mar 29, 2012 at 23:02
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$\begingroup$ Thanks for the helpful sources. 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? $\endgroup$– Jodens PotendsCommented Apr 1, 2012 at 0:12
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$\begingroup$ Well, what book and pages? $\endgroup$– Will JagyCommented Apr 9, 2012 at 19:54
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$\begingroup$ Page 27 of Lectures on Differential Galois Theory by Andy Magid. $\endgroup$– Jodens PotendsCommented Apr 9, 2012 at 20:10
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2 Answers
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Julia Hartmann in 2002 proved the following theorem, which partly answers your question:
Every linear algebraic group defined over the algebraically closed field $K$ occurs as the differential Galois group of some Picard-Vessiot extension of $K(t)$ with derivation $d/dt$.
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Morphism on Physicsforums provided the following answer to this question (I reposted it there). Can someone check this for validity: "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.)"
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1$\begingroup$ Note that the field F is fixed in the question. In Galois theory nomenclatur this would be the much harder (and still unsolved!) inverse Galois problem. $\endgroup$ Commented Jun 4, 2012 at 22:27