A simple proof of the Abel-Ruffini theorem

TL;DR

This paper provides an elementary, accessible proof of the Abel-Ruffini theorem on the unsolvability of general polynomial equations of degree five or higher using radicals, avoiding advanced algebraic concepts.

Contribution

It offers a simplified, expository proof that does not rely on Galois theory, making the theorem more approachable for students and educators.

Findings

Elementary proof of Abel-Ruffini theorem presented

Proof avoids Galois theory terminology

Accessible for students familiar with complex numbers

Abstract

This paper is purely expositional. The statement of the Abel-Ruffini theorem on unsolvability of equations using radicals is simple and well-known. We sketch an elementary proof of this theorem. We do not use the terms 'field extension', 'Galois group' and even 'group'. However, our presentation is a good way to learn (or recall) starting idea of the Galois theory. Our exposition follows `Mathematical Omnibus' of S. Tabachnikov and D.B. Fuchs (in English, http://www.math.psu.edu/tabachni/Books/taba.pdf). The main difference is that we show how the proof could have been invented. The paper is accessible for students familiar with complex numbers, and could be an interesting easy reading for mature mathematicians. The material is presented as a sequence of problems, which is peculiar not only to Zen monasteries but also to advanced mathematical education; most problems are presented with hints or solutions.