%0 Journal Article
%T The inverse problem of differential Galois theory over the field R(z)
%A Tobias Dyckerhoff
%J Mathematics
%D 2008
%I arXiv
%X We describe a Picard-Vessiot theory for differential fields with non algebraically closed fields of constants. As a technique for constructing and classifying Picard-Vessiot extensions, we develop a Galois descent theory. We utilize this theory to prove that every linear algebraic group $G$ over $\mathbb{R}$ occurs as a differential Galois group over $\mathbb{R}(z)$. The main ingredient of the proof is the Riemann-Hilbert correspondence for regular singular differential equations over $\mathbb{C}(z)$.
%U http://arxiv.org/abs/0802.2897v1