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On Kronecker's Solvability Theorem

Yan Pan, Yuzhen Chen

TL;DR

Kronecker's solvability theorem is revisited through a polynomial complete decomposition framework that parallels Dörrie's elementary approach while addressing a gap. The work develops radical-extension machinery, including irreducible radical towers and complex conjugate closure, and proves a complete prime-degree decomposition result to express roots in a controlled radical form. Applying these tools to solvable irreducible polynomials of prime degree with complex-conjugate pairs, the paper shows that only one real root can occur, thereby establishing Kronecker's solvability criterion; it also fixes a gap in Dörrie's argument by carefully handling conjugate radicals. The results clarify the interaction between radical solvability, roots of unity, and complex conjugation, and provide a self-contained derivation of Kronecker's theorem from first principles.

Abstract

Kronecker's 1856 paper contains a solvability theorem that is useful to construct unsolvable algebraic equations. We show how Kronecker's solvability theorem can be derived naturally via a polynomial complete decomposition method. This method is similar to Dörrie, but we fill a gap that appears in his proof.

On Kronecker's Solvability Theorem

TL;DR

Kronecker's solvability theorem is revisited through a polynomial complete decomposition framework that parallels Dörrie's elementary approach while addressing a gap. The work develops radical-extension machinery, including irreducible radical towers and complex conjugate closure, and proves a complete prime-degree decomposition result to express roots in a controlled radical form. Applying these tools to solvable irreducible polynomials of prime degree with complex-conjugate pairs, the paper shows that only one real root can occur, thereby establishing Kronecker's solvability criterion; it also fixes a gap in Dörrie's argument by carefully handling conjugate radicals. The results clarify the interaction between radical solvability, roots of unity, and complex conjugation, and provide a self-contained derivation of Kronecker's theorem from first principles.

Abstract

Kronecker's 1856 paper contains a solvability theorem that is useful to construct unsolvable algebraic equations. We show how Kronecker's solvability theorem can be derived naturally via a polynomial complete decomposition method. This method is similar to Dörrie, but we fill a gap that appears in his proof.

Paper Structure

This paper contains 11 sections, 12 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Some definitions and results on radical extension
  3. An irreducible radical tower with enough roots of unity
  4. A complete decomposition theorem on polynomial of prime degree
  5. An irreducible complex conjugate closed radical tower
  6. Proof of Kronecker's solvability theorem
  7. A supplement to Dörrie's proof
  8. Acknowledgments
  9. Proof of \ref{['eq:KroDor']}
  10. Proof of Theorem \ref{['th:Gauss']}
  11. Proof of Theorem \ref{['th:Gauss']}

Key Result

Proposition 1

If an irreducible polynomial with integer coefficients is solvable and its degree is an odd prime, then either all of its roots or only one of them is real.

Theorems & Definitions (30)

  • Proposition 1: Kronecker
  • Proposition 2: Galois
  • Theorem 1: edwards2014roots, Theorem 3.1's weak version
  • Definition 1
  • Definition 2
  • Lemma 1: Abel
  • proof
  • Corollary 1
  • proof
  • Remark 1
  • ...and 20 more