# Comments on J. F. Ritt's book "Integration in Finite Terms"

## TL;DR

This paper offers modern proofs of Liouville's theorems on integrability and extends them, while also outlining a topological Galois theory approach to solving differential equations in finite terms.

## Contribution

It provides new proofs of classical theorems and introduces a topological Galois theory framework for analyzing solvability of differential equations.

## Key findings

- Modern proofs of Liouville's theorems
- Extension of the second theorem to higher-order equations
- Outline of topological Galois theory and its applications

## Abstract

The First and Second Liouville's Theorems provide correspondingly criterium for integrability of elementary functions "in finite terms" and criterium for solvability of second order linear differential equations by quadratures. The brilliant book of J.F.~Ritt contains proofs of these theorems and many other interesting results. This paper was written as comments on the book but one can read it independently. The first part of the paper contains modern proofs of The First Theorem and of a generalization of the Second Theorem for linear differential equations of any order. In the second part of the paper we present an outline of topological Galois theory which provides an alternative approach to the problem of solvability of equations in finite terms. The first section of this part deals with a topological approach to representability of algebraic functions by radicals and to the 13-th Hilbert problem. This section is written with all proofs. Next sections contain only statements of results and comments on them (basically no proofs are presented there).