Let $a_1,\ldots,a_r\in R$ be elements of a commutative ring and let $x$ be a variable. We form the polynomial. $$\begin{array}{rcl}p(x)&:=&(x-a_1)\cdots (x-a_r)\\ &=&x^r+(-1)^1\Sigma_1x^{r-1}+(-1)^2\Sigma_2x^{r-2}+\ldots+(-1)^{r-1}\Sigma_{r-1}x+(-1)^{r}\Sigma_r \end{array}$$
The coefficients $\Sigma_1,\ldots,\Sigma_r$ are called elementary symmetric functions $\Sigma_k:R^r\to R$, which are defined as sums. $$\begin{array}{rcl} \Sigma_1(a_1,\ldots,a_r)&:=&\sum_{1\le k\le r}a_k\\ \Sigma_2(a_1,\ldots,a_r)&:=&\sum_{1\le k < l\le r}a_ka_l\\ \Sigma_3(a_1,\ldots,a_r)&:=&\sum_{1\le k < l < m\le r}a_ka_la_m\\ \vdots&&\\ \Sigma_r(a_1,\ldots,a_r)&:=&a_1\cdots a_r\\ \end{array}$$
Proofs: 1
