 # Fundamental Theorem on Symmetric Polynomials: History’s First Whiff of Galois Theory

by Ben Blum-Smith and Samuel Coskey

Year of Award: 2018

Award: Pólya

Publication Information: The College Mathematics Journal, Volume 48, Number 1, January 2017, pages 18-29.

Summary:  Modern treatments of Galois theory rely on facts about vector spaces over a field. This material was not available to Galois, whose approach mirrored Newton's explorations of symmetric polynomials. This article focuses on these polynomials as an accessible way for readers who do not know Galois theory to get acquainted, while also providing insight to informed readers that they may have missed in more streamlined treatments of the subject.

Symmetric polynomials (e.g., 4x1x2x3 and x1 + 2x1x2 + x2) are unchanged by any permutation of their variables, and among them are the elementary symmetric polynomials (e.g., in three variables, x1 + x2+ x3, x1x2+x1x3+x2x3 and x1x2x3). The fundamental theorem on symmetric polynomials (FTSP) reads:

Any symmetric polynomial in n variables may be represented in a unique way as a polynomial in the elementary symmetric polynomials in n variables.

For example (from the article), the symmetric polynomial (x1x2)2 can be represented as (x1 + x2)2 − 4x1x2. The authors give two proofs of this result—one classical (by Gauss), the other developed by participants in a 2009 course taught by the first author—and begin to shed light on its connection with Galois Theory. The classical proof uses lexicographic order to identify a leading monomial, which is then eliminated by subtracting an appropriate product of elementary symmetric polynomials. The authors suggest that focusing on a single monomial hides the symmetry that originally interested Newton and Galois. They propose an alternate proof, logically similar to the first proof but first identifying (and then eliminating) a full symmetry class consisting of monomials for which the exponent strings have maximum variance. This proof appeals more directly to symmetry and adds geometric insight via an appealing and intuitive “brick-stacking” argument. The authors then explain how the first proof is, in a sense, the limit of the second, and sketch an argument proving this by extending the original variance proof to arbitrary pth moments.

Response from the Authors:

We are very honored to receive this award. This article grew out of Ben’s desire to explore the teaching and learning of the foundations of Galois theory. He organized a like-minded group of learners and began teaching about symmetries, and later polynomials and solvability. One day a lesson about the fundamental theorem on symmetric polynomials hit an obstacle because the participants could not think of the clever trick needed to continue. Instead of accepting the solution and moving on, the two of us searched for another approach that would not have seemed so clever. Along the way, we explored some history as well as some new ways to look at the problem. Any small investigation between friends or students, together with some follow-up sleuthing, can lead to an article just like this one. We are grateful to the CMJ for providing a forum for this type of exposition.

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