Thus, when working only with such symmetric polynomial expressions in the roots, it is unnecessary to know anything particular about those roots, or to compute in any larger field than K in which those roots may lie. In fact the values of the roots themselves become rather irrelevant, and the necessary relations between coefficients and symmetric polynomial expressions can be found by computations in terms of symmetric polynomials only. An example of such relations are Newton's identities, which express the sum of any fixed power of the roots in terms of the elementary symmetric polynomials. Special kinds of symmetric polynomials There are a few types of symmetric polynomials in the variables X1, X2, …, Xn that are fundamental. Elementary symmetric polynomials For each nonnegative integer k, the elementary symmetric polynomial ek(X1, …, Xn) is the sum of all distinct products of k distinct variables. (Some authors denote it by σk instead.) For k = 0 there is only the empty product so e0(X1, …, Xn) = 1, while for k > n, no products at all can be formed, so ek(X1, X2, …, Xn) = 0 in these cases. The remaining n elementary symmetric polynomials are building blocks for all symmetric polynomials in these variables: as mentioned above, any symmetric polynomial in the variables considered can be obtained from these elementary symmetric polynomials using multiplications and additions only. In fact one has the following more detailed facts: any symmetric polynomial P in X1, …, Xn can be written as a polynomial expression in the polynomials ek(X1, …, Xn) with 1 ≤ k ≤ n; this expression is unique up to equivalence of polynomial expressions; if P has integral coefficients, then the polynomial expression also has integral coefficients. For example, for n = 2, the relevant elementary symmetric polynomials are e1(X1, X2) = X1 + X2, and e2(X1, X2) = X1X2. The first polynomial in the list of examples above can then be written as (for a proof that this is always possible see the fundamental theorem of symmetric polynomials). Monomial symmetric polynomials Powers and products of elementary symmetric polynomials work out to rather complicated expressions. If one seeks basic additive building blocks for symmetric polynomials, a more natural choice is to take those symmetric polynomials that contain only one type of monomial, with only those copies required to obtain symmetry. Any monomial in X1, …, Xn can be written as