The easiest way to see that it holds is from an identity of formal power series in for the elementary symmetric polynomials, analogous to the one given above for the complete homogeneous ones:
22.
As the discriminant is a symmetric function in the roots, it can also be expressed in terms of the coefficients of the polynomial, since the coefficients are the elementary symmetric polynomials in the roots.
23.
Grouping these terms by degree yields the elementary symmetric polynomials in x _ i for " x k, " all distinct " k "-fold products of x _ i.
24.
Using them in reverse to express the elementary symmetric polynomials in terms of the power sums, they can be used to find the characteristic polynomial by computing only the powers " A " " k " and their traces.
25.
As the permutation group of the roots is generated by these permutations, it follows that and are symmetric functions of the roots and may thus be written as polynomials in the elementary symmetric polynomials and thus as rational functions of the coefficients of the equation.
26.
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.
27.
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.
28.
A basic fact, known as the "'fundamental theorem of symmetric polynomials "'states that " any " symmetric polynomial in " n " variables can be given by a polynomial expression in terms of these elementary symmetric polynomials.
29.
A theorem states that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials, which implies that every " symmetric " polynomial expression in the roots of a monic polynomial can alternatively be given as a polynomial expression in the coefficients of the polynomial.
30.
When we substitute these eigenvalues into the elementary symmetric polynomials, we obtain, up to their sign, the coefficients of the characteristic polynomial, which are trace ( the sum of the elements of the diagonal ) is the value of, and thus the sum of the eigenvalues.
