Editing Symmetric polynomial (section)

== Applications ==

=== Galois theory ===
{{Main|Galois theory}}
One context in which symmetric polynomial functions occur is in the study of [[Monic polynomial|monic]] [[univariate]] polynomials of [[Degree of a polynomial|degree]] ''n'' having ''n'' roots in a given [[field (mathematics)|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 by inverting this map, "breaking" the symmetry – given the coefficients of the polynomial (the [[elementary symmetric polynomial]]s in the roots), how can one recover the roots?
This leads to studying solutions of polynomials using the [[permutation group]] of the roots, originally in the form of [[Lagrange resolvents]], later developed in [[Galois theory]].