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 e1 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. Complete homogeneous symmetric polynomials For each nonnegative integer k, the complete homogeneous symmetric polynomial hk(X1, …, Xn) is the sum of all distinct monomials of degree k in the variables X1, …, Xn. For instance The polynomial hk(X1, …, Xn) is also the sum of all distinct monomial symmetric polynomials of degree k in X1, …, Xn, for instance for the given example All symmetric polynomials in these variables can be built up from complete homogeneous ones: any symmetric polynomial in X1, …, Xn can be obtained from the complete homogeneous symmetric polynomials h1(X1, …, Xn), …, hn(X1, …, Xn) via multiplications and additions. More precisely: Any symmetric polynomial P in X1, …, Xn can be written as a polynomial expression in the polynomials hk(X1, …, Xn) with 1 ≤ k ≤ n. If P has integral coefficients, then the polynomial expression also has integral coefficients. For example, for n = 2, the relevant complete homogeneous symmetric polynomials are and . The first polynomial in the list of examples above can then be written as As in the case of power sums, the given statement applies in particular to the complete homogeneous symmetric polynomials beyond hn(X1, …, Xn), allowing them to be expressed in terms of the ones up to that point; again the resulting identities become invalid when the number of variables is increased. An important aspect of complete homogeneous symmetric polynomials is their relation to elementary symmetric polynomials, which can be expressed as the identities , for all k > 0, and any number of variables n. Since e0(X1, …, Xn) and h0(X1, …, Xn) are both equal to 1, one can isolate either the first or the last term of these summations; the former gives a set of equations that allows one to recursively express the successive complete homogeneous