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### Abstract

A symmetric polynomial is a polynomial in variables
*X*_{1},…,*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
e_{1},…,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*x*_{1},…,*x*_{n}, namely*c*_{k}= (-1)^{n-k}e_{n-k}(*x*_{1},…,*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.