turkmath.org

Galois Theory is a powerful tool to study the roots of a polynomial. In this sense, the differential Galois theory is the analogue of Galois theory for linear differential equations. In this talk, we will construct the notion of a differential field and PicardVessiot 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 corollay 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 in terms of elementary functions.