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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $n \\ge 1$.\nA polynomial $f \\in R[x_1, \\dots, x_n]$ is symmetric if\n$$f(x_{\\sigma(1)}, \\dots, x_{\\sigma(n)}) = f(x_1, \\dots, x_n)$$\nfor every permutation $\\sigma$ of $\\{1, \\dots, n\\}$.\n\n## Remarks\n\nExamples in two variables: $x_1 + x_2$, $x_1 x_2$ and $x_1^2 + x_2^2$. The polynomial $x_1 - x_2$ is not symmetric.\nThe symmetric polynomials form a subring of $R[x_1, \\dots, x_n]$, the ring of invariants of the symmetric group.\nThe main examples are the [[elementary symmetric polynomials|0ZMY8441GW]] and the [[power sums|0AS1FZT6BM]].\nThe coefficients of a monic polynomial are symmetric polynomials in its roots, by [[Vieta's formulas|0M8EWQJ1YV]].\n\n## Explanations\n\n### L1 human\n\nA symmetric polynomial treats all its variables in the same way.\nSwapping two variables gives the same polynomial.\nThis is why symmetric expressions in the roots of a polynomial can be computed from the coefficients.\n",
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