A double demonstration of a theorem of Newton, which gives a relation between the coefficient of an algebraic equation and the sums of the powers of its roots

2007/07/04 by Leonhard Euler, Euler, Leonhard
Mathematics · #01A50 #12D10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #General Mathematics (math.GM) #History and Overview (math.HO) #History and Theory of Mathematics #Mathematics and Applications #math.CA #math.GM #math.HO #msc:01A50 #msc:12D10

paper · pdf · doi:10.48550/arxiv.0707.0699

9 pages

arxiv created 2007/07/04 · openalex publication_date 2007/07/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Translation from the Latin original, "Demonstratio gemina theorematis Neutoniani, quo traditur relatio inter coefficientes cuiusvis aequationis algebraicae et summas potestatum radicum eiusdem" (1747). E153 in the Enestrom index. In this paper Euler gives two proofs of Newton's identities, which express the sums of powers of the roots of a polynomial in terms of its coefficients. The first proof takes the derivative of a logarithm. The second proof uses induction and the fact that in a polynomial of degree n, the coefficient of x^n-k is equal to the sum of the products of k roots, times (-1)^k.

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