Abel-Ruffini's Theorem: Complex but Not Complicated!

TL;DR

This paper provides a simple, elementary proof of Abel-Ruffini's theorem, explaining why algebraic solutions using radicals exist for degree four or less but not for degree five or more, accessible to students without advanced algebra.

Contribution

It offers a self-contained, elementary proof of Abel-Ruffini's theorem, avoiding complex Galois theory, making the result more accessible to learners.

Findings

Elementary proof of Abel-Ruffini's theorem presented

Clarifies why solutions exist for degree four or less

Shows why no radical solutions exist for degree five or more

Abstract

In this article, using only elementary knowledge of complex numbers, we sketch a proof of the celebrated Abel--Ruffini theorem, which states that the general solution to an algebraic equation of degree five or more cannot be written using radicals, that is, using its coefficients and arithmetic operations $+,-,\times,\div,$ and $\sqrt{\ }$. The present article is written purposely with concise and pedagogical terms and dedicated to students and researchers not familiar with Galois theory, or even group theory in general, which are the usual tools used to prove this remarkable theorem. In particular, the proof is self-contained and gives some insight as to why formulae exist for equations of degree four or less (and how they are constructed), and why they do not for degree five or more.