The Fundamental Theorem on Symmetric Polynomials: History's First Whiff of Galois Theory

TL;DR

This paper explores the Fundamental Theorem on Symmetric Polynomials, providing both classical and novel proofs, and discusses its historical significance and connection to Galois theory and symmetry in algebra.

Contribution

It introduces a new proof of the FTSP derived from group theory, contextualized within its historical and pedagogical development.

Findings

Provides a classical proof of FTSP

Introduces a novel proof from group theory

Discusses FTSP's role in Galois theory and symmetry

Abstract

We describe the Fundamental Theorem on Symmetric Polynomials (FTSP), exposit a classical proof, and offer a novel proof that arose out of an informal course on group theory. The paper develops this proof in tandem with the pedagogical context that led to it. We also discuss the role of the FTSP both as a lemma in the original historical development of Galois theory and as an early example of the connection between symmetry and expressibility that is described by the theory.