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 is given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials. There is one elementary symmetric polynomial of degree in variables for each positive integer , and it is formed by adding together all distinct products of distinct variables. Definition The elementary symmetric polynomials in variables , written for , are defined by and so forth, ending with In general, for we define so that if . (Sometimes, is included among the elementary symmetric polynomials, but excluding it allows generally simpler formulation of results and properties.) Thus, for each positive integer less than or equal to there exists exactly one elementary symmetric polynomial of degree in variables. To form the one that has degree , we take the sum of all products of -subsets of the 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) , one defines the symmetric polynomial , also called an elementary symmetric polynomial, by . Sometimes the notation is used instead of . Examples The following lists the elementary symmetric polynomials for the first four positive values of . For : For : For : For : Properties 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 , we obtain the monic univariate polynomial (with variable ) whose roots are the values substituted for and whose coefficients are – up to their sign – the elementary symmetric polynomials. These relations between the roots and the coefficients of a polynomial are called Vieta's formulas. The characteristic polynomial of a square matrix is an example of application of Vieta's formulas. The roots of this polynomial are the eigenvalues of the matrix. When we substitute these eigenvalues into the elementary symmetric polynomials, we obtain – up to their sign – the coefficients of the characteristic polynomial, which are invariants of the matrix. In particular, the trace (the sum of the