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  "body": "\n## Statement\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $f \\in R[x_1, \\dots, x_n]$ be [[symmetric|06CTFY2DEZ]].\nThen there is a unique polynomial $g \\in R[y_1, \\dots, y_n]$ with\n$$f = g(e_1, \\dots, e_n),$$\nwhere $e_1, \\dots, e_n$ are the [[elementary symmetric polynomials|0ZMY8441GW]].\n\n## Remarks\n\nEquivalently, the map $R[y_1, \\dots, y_n] \\to R[x_1, \\dots, x_n]$, $y_k \\mapsto e_k$, is injective, and its image is the ring of symmetric polynomials.\nOne proof is by induction on the leading monomial in the lexicographic order. Each step subtracts a product of the $e_k$ with the same leading monomial.\nExample: $x_1^2 + x_2^2 = e_1^2 - 2 e_2$.\nSo a symmetric expression in the roots of a polynomial over a field $K$ is an element of $K$.\n\n## Explanations\n\n### L1 human\n\nEvery symmetric polynomial can be written in exactly one way using the elementary symmetric polynomials.\nThe coefficients stay in the same ring.\n",
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