Galois Theory
One context in which symmetric polynomial functions occur is in the study of monic univariate polynomials of degree n having n roots in a given field. These n roots determine the polynomial, and when they are considered as independent variables, the coefficients of the polynomial are symmetric polynomial functions of the roots. Moreover the fundamental theorem of symmetric polynomials implies that a polynomial function f of the n roots can be expressed as (another) polynomial function of the coefficients of the polynomial determined by the roots if and only if f is given by a symmetric polynomial.
This yields the approach to solving polynomial equations in terms of inverting this map, "breaking" the symmetry – given the coefficients of the polynomial (the elementary symmetric polynomials in the roots), how can one recover the roots? This leads to studying solutions of polynomials in terms of the permutation group of the roots, originally in the form of Lagrange resolvents, later developed in Galois theory.
Read more about this topic: Symmetric Polynomial, Applications
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