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  "body": "\n## Statement\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $n \\ge 1$.\nLet $e_k$ be the [[elementary symmetric polynomials|0ZMY8441GW]] and $p_k$ the [[power sums|0AS1FZT6BM]] in $x_1, \\dots, x_n$, with $e_0 = 1$ and $e_k = 0$ for $k \u003e n$.\nThen for every $k \\ge 1$,\n$$k \\, e_k = \\sum_{i=1}^{k} (-1)^{i-1} e_{k-i} \\, p_i.$$\n\n## Remarks\n\nThe first cases are $e_1 = p_1$, $2 e_2 = e_1 p_1 - p_2$ and $3 e_3 = e_2 p_1 - e_1 p_2 + p_3$.\nFor $k \u003e n$, the left side is $0$, and the identity expresses $p_k$ through $p_{k-1}, \\dots, p_{k-n}$.\nApplied to the roots of a polynomial, the identities compute the power sums of the roots from the coefficients.\nApplied to the eigenvalues of a matrix $A$, they relate the traces of the powers of $A$ to the coefficients of its characteristic polynomial.\n\n## History\n\nIsaac Newton found the identities around 1666.\nAlbert Girard had already found the first four formulas in 1629.\n\n## Explanations\n\n### L1 human\n\nNewton's identities convert between two ways of describing the roots of a polynomial: the sums of their powers, and the coefficients.\nEach identity gives the next quantity from the earlier ones.\n",
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