A short elementary proof of the insolvability of the equation of degree 5

TL;DR

This paper offers concise, elementary proofs of the classical results on the insolvability of degree 5 polynomial equations, connecting basic algebra with Galois theory without advanced group concepts.

Contribution

It provides simplified proofs of insolvability theorems that avoid complex group theory, making the concepts more accessible and bridging elementary algebra with Galois theory.

Findings

Elementary proofs of insolvability of degree 5 equations

Avoidance of Galois group terminology in proofs

Accessible to students with basic polynomial and permutation knowledge

Abstract

We present short elementary proofs of the well-known Ruffini-Abel-Galois theorems on insolvability of algebraic equations in radicals. These proofs are obtained from existing expositions by stripping away material not required for the proofs (but presumably required elsewhere). In particular, we do not use the terms `Galois group' and even `group'. However, our presentation is a good way to learn (or to recall) a starting idea of Galois theory: the symmetry of a polynomial of several variables is decreased when a radical is extracted. So the note provides a bridge (by showing that there is no gap) between elementary mathematics and Galois theory. The note is accessible to students familiar with polynomials, complex numbers and permutations; so the note might be interesting easy reading for professional mathematicians.