We give here an algorithm for reducing a symmetric polynomial into a polynomial in the elementary symmetric polynomials.

We define the height of a monomial $x_1^{e_1} \cdots x_n^{e_n}$ in $R[x_1,\dots,x_n]$ to be $e_1 + 2 e_2 + \cdots + n e_n$ The height of a polynomial is defined to be the maximum height of any of its monomial terms, or 0 if it is the zero polynomial.

Let $f$ be a symmetric polynomial. We reduce $f$ into elementary symmetric polynomials by induction on the height of $f$ Let $c x_1^{e_1} \cdots x_n^{e_n}$ be the monomial term of maximal height in $f$ Consider the polynomial $$ g := f - c s_1^{e_n - e_{n-1}} s_2^{e_{n-1} - e_{n-2}} \cdots s_{n-1}^{e_2 - e_1} s_n^{e_1} $$ where $s_k$ is the $k$ -th elementary symmetric polynomial in the $n$ variables $x_1, \dots, x_n$ Then $g$ is a symmetric polynomial of lower height than $f$ so by the induction hypothesis, $g$ is a polynomial in $s_1, \dots, s_n$ and it follows immediately that $f$ is also a polynomial in $s_1, \dots, s_n$