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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $n \\ge 1$.\nFor $k \\ge 1$, the $k$-th power sum in $x_1, \\dots, x_n$ is\n$$p_k = x_1^k + x_2^k + \\dots + x_n^k \\in R[x_1, \\dots, x_n].$$\n\n## Remarks\n\nEach $p_k$ is [[symmetric|06CTFY2DEZ]] and homogeneous of degree $k$.\n[[Newton's identities|0WG2A3HE0S]] relate the power sums to the [[elementary symmetric polynomials|0ZMY8441GW]].\nIf $R$ contains $\\mathbb{Q}$, then every symmetric polynomial in $R[x_1, \\dots, x_n]$ is a polynomial in $p_1, \\dots, p_n$.\nThis fails over $\\mathbb{Z}$: $e_2 = \\tfrac{1}{2}(p_1^2 - p_2)$ needs the coefficient $\\tfrac{1}{2}$.\n\n## Explanations\n\n### L1 human\n\nThe power sums add up the same power of every variable.\nThey are symmetric, and over the rationals they can replace the elementary symmetric polynomials as generators.\n",
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