Symmetric Polynomials

A symmetric polynomial is a polynomial in variables X₁,…,X_n that does not discriminate between its variables, i. e. it is invariant under any permutation of them. These polynomials are important in the study of the relationship between the coefficients of a univariate polynomial and its roots in its algebraic closure.

This article provides a definition of symmetric polynomials and the elementary symmetric polynomials e₁,…,e_n and proofs of their basic properties, including three notable ones:

• Vieta's formula, which gives an explicit expression for the k-th coefficient of a univariate monic polynomial in terms of its roots x₁,…,x_n, namely c_k = (-1)^n-k e_n-k(x₁,…,x_n).
• Second, the Fundamental Theorem of Symmetric Polynomials, which states that any symmetric polynomial is itself a uniquely determined polynomial combination of the elementary symmetric polynomials.
• Third, as a corollary of the previous two, that given a polynomial over some ring R, any symmetric polynomial combination of its roots is also in R even when the roots are not.

Both the symmetry property itself and the witness for the Fundamental Theorem are executable.