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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $n \\ge 1$.\nFor $0 \\le k \\le n$, the $k$-th elementary symmetric polynomial in $x_1, \\dots, x_n$ is\n$$e_k = \\sum_{1 \\le i_1 \u003c i_2 \u003c \\dots \u003c i_k \\le n} x_{i_1} x_{i_2} \\cdots x_{i_k} \\in R[x_1, \\dots, x_n].$$\n\n## Remarks\n\nSo $e_0 = 1$, $e_1 = x_1 + \\dots + x_n$ and $e_n = x_1 x_2 \\cdots x_n$. One sets $e_k = 0$ for $k \u003e n$.\nIn three variables, $e_2 = x_1 x_2 + x_1 x_3 + x_2 x_3$.\nThe generating identity is $\\prod_{i=1}^n (t + x_i) = \\sum_{k=0}^n e_k t^{n-k}$.\nEach $e_k$ is [[symmetric|06CTFY2DEZ]] and homogeneous of degree $k$.\n\n## Explanations\n\n### L1 human\n\nThe elementary symmetric polynomials are the simplest symmetric polynomials: the sum of the variables, the sum of their products in pairs, and so on.\nUp to sign, they are the coefficients of the polynomial whose roots are the variables.\n",
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