On Kronecker's Solvability Theorem

TL;DR

This paper revisits Kronecker's 1856 solvability theorem, providing a natural derivation using a polynomial complete decomposition method and addressing gaps in previous proofs.

Contribution

It introduces a new derivation of Kronecker's solvability theorem through a polynomial decomposition method, improving upon D"orrie's approach.

Findings

Provides a complete and gap-free proof of Kronecker's solvability theorem

Demonstrates the method's effectiveness in constructing unsolvable algebraic equations

Enhances understanding of polynomial decomposition techniques

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 D\"orrie, but we fill a gap that appears in his proof.