Integrability in finite terms and actions of Lie groups

According to Liouville's Theorem, an idefinite integral of an elementary function is usually not an elementary function. In these notes, we discuss that statement and a proof of this result. The differential Galois group of the extension obtained by adjoining an integral does not determine whether the integral is an elementary function or not. Nevertheless, Liouville's Theorem can be proved using differential Galois groups. The first step towards such a proof was suggested by Abel. This step is related to algebraic extensions and their finite Galois groups. A significant part of these notes is dedicated to the second step dealing with pure transcendent extensions and their Galois groups, which are connected Lie groups. The idea of the proof goes back to J. Liouville and J. F. Ritt.