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

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.
Comment: 16 pages, 1 figure. Added an appendix acknowledging a precedent for the "brick-stacking" argument for Theorem 3