Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On unsolvable equations of prime degree

Juliusz Brzeziński, Jan Stevens

TL;DR

We address Kronecker's theorem on solvability by radicals for irreducible polynomials of odd prime degree $p$ over real (conjugation-invariant) fields. An elementary proof is developed by analyzing chains of radical extensions and binomial adjunctions, leveraging Nagell's Lemma and case splits on the irreducibility of $X^q-a$ to establish the real-root dichotomy. The work also scrutinizes Weber's classical elementary approach, identifies a crucial mistake involving reducible radicals, and provides corrected arguments that treat both irreducible and reducible binomials within a unified framework. These results clarify historical developments, sharpen elementary methods, and illustrate how to construct unsolvable-by-radicals polynomials such as $X^5-4X-2$.

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.

On unsolvable equations of prime degree

TL;DR

We address Kronecker's theorem on solvability by radicals for irreducible polynomials of odd prime degree over real (conjugation-invariant) fields. An elementary proof is developed by analyzing chains of radical extensions and binomial adjunctions, leveraging Nagell's Lemma and case splits on the irreducibility of to establish the real-root dichotomy. The work also scrutinizes Weber's classical elementary approach, identifies a crucial mistake involving reducible radicals, and provides corrected arguments that treat both irreducible and reducible binomials within a unified framework. These results clarify historical developments, sharpen elementary methods, and illustrate how to construct unsolvable-by-radicals polynomials such as .

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.
Paper Structure (7 sections, 10 theorems, 12 equations, 1 figure)

This paper contains 7 sections, 10 theorems, 12 equations, 1 figure.

Table of Contents

  1. History.
  2. Preliminaries
  3. What is an elementary proof?
  4. Field extensions.
  5. Auxiliary results.
  6. A proof of Kronecker's Theorem.
  7. Weber's proof.

Key Result

Theorem 1

Let $K$ be a real field. Suppose that an irreducible polynomial $f(X)\in K[X]$ of odd prime degree $p$ is solvable by radicals. Then exactly one root or all roots of the polynomial are real.

Figures (1)

  • Figure 1: The graph of $f(X)=X^5-4X-2$

Theorems & Definitions (16)

  • Theorem 1: Kronecker's Theorem
  • Theorem 2: Galois' Theorem
  • Lemma 3
  • Lemma 4
  • Lemma 5: Nagell's Lemma
  • Lemma 6
  • Lemma 7
  • proof
  • Remark 8
  • Theorem 9: Kronecker's Theorem
  • ...and 6 more