elements of the diagonal) is the value of , and thus the sum of the eigenvalues. Similarly, the determinant is – up to the sign – the constant term of the characteristic polynomial, i.e. the value of . Thus the determinant of a square matrix is the product of the eigenvalues. The set of elementary symmetric polynomials in variables generates the ring of symmetric polynomials in variables. More specifically, the ring of symmetric polynomials with integer coefficients equals the integral polynomial ring . (See below for a more general statement and proof.) This fact is one of the foundations of invariant theory. For another system of symmetric polynomials with the same property see Complete homogeneous symmetric polynomials, and for a system with a similar, but slightly weaker, property see Power sum symmetric polynomial. Fundamental theorem of symmetric polynomials For any commutative ring , denote the ring of symmetric polynomials in the variables with coefficients in by . This is a polynomial ring in the n elementary symmetric polynomials for . This means that every symmetric polynomial has a unique representation for some polynomial . Another way of saying the same thing is that the ring homomorphism that sends to for defines an isomorphism between and . Proof sketch The theorem may be proved for symmetric homogeneous polynomials by a double induction with respect to the number of variables and, for fixed , with respect to the degree of the homogeneous polynomial. The general case then follows by splitting an arbitrary symmetric polynomial into its homogeneous components (which are again symmetric). In the case the result is trivial because every polynomial in one variable is automatically symmetric. Assume now that the theorem has been proved for all polynomials for variables and all symmetric polynomials in variables with degree . Every homogeneous symmetric polynomial in can be decomposed as a sum of homogeneous symmetric polynomials Here the "lacunary part" is defined as the sum of all monomials in which contain only a proper subset of the variables , i.e., where at least one variable is missing. Because is symmetric, the lacunary part is determined by its terms containing only the variables , i.e., which do not contain . More precisely: If and are two homogeneous symmetric polynomials in having the same degree, and if the coefficient of before each monomial which contains only the variables equals the corresponding coefficient of , then