Some more proofs from the Book: solvability and insolvability of equations in radicals

TL;DR

This paper provides simple, elementary proofs of key results in algebra concerning the solvability of equations in radicals, avoiding advanced concepts like Galois groups, making the proofs more accessible.

Contribution

It offers concise, elementary proofs of classical theorems on constructibility and solvability of polynomial equations, without using Galois theory or group terminology.

Findings

Gauss's theorem on constructibility of regular polygons

Existence of a cubic equation unsolvable in real radicals

Existence of a quintic equation unsolvable in complex radicals

Abstract

This paper is purely expository. We present short elementary proofs of * the Gauss Theorem on constructibility of regular polygons; * the existence of a cubic equation unsolvable in real radicals; * the existence of a quintic equation unsolvable in complex radicals (Galois Theorem). The statements of these celebrated results are simple and well-known. However, their proofs given in most textbooks rely upon much unmotivated material and are far from being economic. We do not use the terms `Galois group' or even `group'. The paper is accessible for students familiar with polynomials and complex numbers, and could be an interesting easy reading for professional mathematicians. Short English version is followed by an extended Russian version where before presenting the proofs we illustrate the main ideas by sequences of problems with hints or solutions.