Diferensiyel Galois teorisi

Galois Theory is a powerful tool to study the roots of polynomials. In this sense, the differential Galois theory is the analogue of Galois theory for linear differential equations. In this thesis, we will construct the notion of a differential field and Picard-Vessiot extension of a linear differential equation as the analogue of a field and the splitting field of a polynomial, respectively. Then we define the differential Galois group and we see that it has a linear algebraic group structure. Using those, we have a Galois correspondence for algebraic subgroups of the differential Galois group similar to the correspondence in the Galois theory. Moreover, we find a characterization for Liouvillian functions corresponding to the solvability of $G^0$ , the identity component of differential Galois group $G$. This is the analogue of the characterization of solvability by radicals of a polynomial equation in Galois theory. As a corollary we find that identity component of the differential Galois group of an elementary function is abelian. Using this tool we can prove that $\int e^{-x^2}$ cannot be expressed as an elementary function. Besides, there is a connection between differential Galois theory and Tannakian categories. We also present this approach.