Elementary symmetric polynomials

Definition

Let $R$ be a commutative ring and let $n \ge 1$. For $0 \le k \le n$, the $k$-th elementary symmetric polynomial in $x_1, \dots, x_n$ is $$e_k = \sum_{1 \le i_1 < i_2 < \dots < i_k \le n} x_{i_1} x_{i_2} \cdots x_{i_k} \in R[x_1, \dots, x_n].$$

Remarks

So $e_0 = 1$, $e_1 = x_1 + \dots + x_n$ and $e_n = x_1 x_2 \cdots x_n$. One sets $e_k = 0$ for $k > n$. In three variables, $e_2 = x_1 x_2 + x_1 x_3 + x_2 x_3$. The generating identity is $\prod_{i=1}^n (t + x_i) = \sum_{k=0}^n e_k t^{n-k}$. Each $e_k$ is symmetric and homogeneous of degree $k$.

Explanations

L1 human

The elementary symmetric polynomials are the simplest symmetric polynomials: the sum of the variables, the sum of their products in pairs, and so on. Up to sign, they are the coefficients of the polynomial whose roots are the variables.