On unsolvable equations of prime degree

TL;DR

This paper provides a concise proof of Kronecker's theorem on the roots of irreducible equations of prime degree, discusses errors in Weber's proof, and offers corrections, contributing to the understanding of unsolvable prime degree equations.

Contribution

It presents a new, shorter proof of Kronecker's theorem and clarifies errors in Weber's original proof, enhancing the theoretical understanding of algebraic equations of prime degree.

Findings

A short proof of Kronecker's theorem is provided.

Identification and correction of errors in Weber's proof.

Insights into the nature of roots of prime degree equations.

Abstract

Kronecker observed that either all roots or only one root of a solvable irreducible equation of odd prime degree with integer coefficients are real. This gives a possibility to construct specific examples of equations not solvable by radicals. A relatively elementary proof without using the full power of Galois theory is due to Weber. We give a rather short proof of Kronecker's theorem with a slightly different argument from Weber's. Several modern presentations of Weber's proof contain inaccuracies, which can be traced back to an error in the original proof. We discuss this error and how it can be corrected.