# On Representability of Algebraic Functions by Radicals

## TL;DR

This paper provides a simplified proof of classical criteria for representing algebraic functions by radicals, extending to algebroidal functions and relating to the 13th Hilbert problem, with historical context and new insights.

## Contribution

It offers a self-contained, simplified proof of classical criteria for algebraic functions' representability by radicals and extends these criteria to algebroidal functions and the 13th Hilbert problem.

## Key findings

- Simplified proof of classical criteria for algebraic functions by radicals
- Criteria for algebroidal functions by composition of analytic functions and radicals
- Connection to the 13th Hilbert problem

## Abstract

This preprint is dedicated to a self contained simple proof of the classical criteria for representability of algebraic functions of several complex variables by radicals. It also contains a criteria for representability of algebroidal functions by composition of single-valued analytic functions and radicals, and a result related to the 13-th Hilbert problem. This preprint is an extended version of the author's 1971 paper. It is written as a part of the comments to a new edition (in preparation) of the classical book `Integration in finite terms' by J.F.~Ritt.