Pinar Larsson: When Differential Equations meet Galois Theory

Analogous to the classical Galois theory which provides an important link between field extensions and subgroups of the symmetric group, differential Galois theory relates differential field extensions and subgroups of the general linear group. In this thesis we introduce differential extensions and construct the Picard-Vessiot extension field L for given base field K and a differential linear homogeneous equation. Furthermore we examine the connection between differential K-automorphisms of L to GL(V), where V stands for the vector space of solutions of the given linear homogeneous differential equation over the field of constants of K, which in its turn is equal to the field of constants of L. Moreover we are examining the algebraic group structure of differential K-automorphisms of Picard-Vessiot extension fields. Also, Liouville and generalized Liouville extensions are examined and their connections to the differential Galois groups are presented.