In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary symmetric polynomials. That is, any symmetric polynomial P is given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials. There is one elementary symmetric polynomial of degree d in n variables for each positive integer d ≤ n, and it is formed by adding together all distinct products of d distinct variables. The elementary symmetric polynomials in n variables X1, ..., Xn, written ek(X1, ..., Xn) for k = 1, ..., n, are defined by and so forth, ending with In general, for k ≥ 0 we define so that ek(X1, ..., Xn) = 0 if k > n. (Sometimes, 1 = e0(X1, ..., Xn) is included among the elementary symmetric polynomials, but excluding it allows generally simpler formulation of results and properties.) Thus, for each positive integer k less than or equal to n there exists exactly one elementary symmetric polynomial of degree k in n variables. To form the one that has degree k, we take the sum of all products of k-subsets of the n variables. (By contrast, if one performs the same operation using multisets of variables, that is, taking variables with repetition, one arrives at the complete homogeneous symmetric polynomials.) Given an integer partition (that is, a finite non-increasing sequence of positive integers) λ = (λ1, ..., λm), one defines the symmetric polynomial eλ(X1, ..., Xn), also called an elementary symmetric polynomial, by Sometimes the notation σk is used instead of ek. The following lists the n elementary symmetric polynomials for the first four positive values of n. For n = 1: For n = 2: For n = 3: For n = 4: The elementary symmetric polynomials appear when we expand a linear factorization of a monic polynomial: we have the identity That is, when we substitute numerical values for the variables X1, X2, ...