---
tag: 0ZMY8441GW
kind: definition
name: Elementary symmetric polynomials
msc: ["05E05"]
defines: 0X5QVCF06N
uses: [06CTFY2DEZ, 0CES91M79A]
sources:
  - url: https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial
    locator: Definition
---

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] 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|06CTFY2DEZ]] 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.