**Power-sum Symmetric Polynomials**

For each integer *k* ≥ 1, the monomial symmetric polynomial *m*_{(k,0,…,0)}(*X*_{1}, …, *X*_{n}) is of special interest, and called the power sum symmetric polynomial *p*_{k}(*X*_{1}, …, *X*_{n}), so

All symmetric polynomials can be obtained from the first *n* power sum symmetric polynomials by additions and multiplications, possibly involving rational coefficients. More precisely,

- Any symmetric polynomial in
*X*_{1}, …,*X*_{n}can be expressed as a polynomial expression with rational coefficients in the power sum symmetric polynomials*p*_{1}(*X*_{1}, …,*X*_{n}), …,*p*_{n}(*X*_{1}, …,*X*_{n}).

In particular, the remaining power sum polynomials *p*_{k}(*X*_{1}, …, *X*_{n}) for *k* > *n* can be so expressed in the first *n* power sum polynomials; for example

In contrast to the situation for the elementary and complete homogeneous polynomials, a symmetric polynomial in *n* variables with *integral* coefficients need not be a polynomial function with integral coefficients of the power sum symmetric polynomials. For an example, for *n* = 2, the symmetric polynomial

has the expression

Using three variables one gets a different expression

The corresponding expression was valid for two variables as well (it suffices to set *X*_{3} to zero), but since it involves *p*_{3}, it could not be used to illustrate the statement for *n* = 2. The example shows that whether or not the expression for a given monomial symmetric polynomial in terms of the first *n* power sum polynomials involves rational coefficients may depend on *n*. But rational coefficients are *always* needed to express elementary symmetric polynomials (except the constant ones, and *e*_{1} which coincides with the first power sum) in terms of power sum polynomials. The Newton identities provide an explicit method to do this; it involves division by integers up to *n*, which explains the rational coefficients. Because of these divisions, the mentioned statement fails in general when coefficients are taken in a field of finite characteristic; however it is valid with coefficients in any ring containing the rational numbers.

Read more about this topic: Symmetric Polynomial, Special Kinds of Symmetric Polynomials