Newton's Identities give a recursive relation between the power sums of the roots of a polynomial, meaning the sums of each root raised to successive whole-number powers, and the elementary symmetric polynomials in those same roots, which up to sign are exactly the polynomial's own coefficients. Named for Isaac Newton, they let either family of quantities be computed from the other without ever finding the roots themselves, and are used throughout algebra and combinatorics.
Facts
StatementNewton's identities relate power sums of the roots of a monic polynomial to its elementary symmetric polynomials, expressing the sums of k-th powers of the roots in terms of the coefficients without finding the roots. 1 Classification
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Source Newton's identities, Wikipedia
Source Newton's identities, Wikipedia
Source Newton's identities, Wikipedia
Named After
Principal Result Of
Source Wikipedia: Isaac Newton
Proved By
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1. Newton's identities, Wikipedia
Lead, second sentence
Evaluated at the roots of a monic polynomial P in one variable, they allow expressing the sums of the k-th powers of all roots of P (counted with their multiplicity) in terms of the coefficients of P, without actually finding those roots.
History section, discovery sentence
These identities were found by Isaac Newton around 1666, apparently in ignorance of earlier work (1629) by Albert Girard.
In Branch: Galois Theory, Lead paragraph
They have applications in many areas of mathematics, including Galois theory, invariant theory, group theory, combinatorics, as well as further applications outside mathematics, including general relativity.
In Branch: Group Theory, Lead paragraph
They have applications in many areas of mathematics, including Galois theory, invariant theory, group theory, combinatorics, as well as further applications outside mathematics, including general relativity.
In Branch: Combinatorics, Lead paragraph
They have applications in many areas of mathematics, including Galois theory, invariant theory, group theory, combinatorics, as well as further applications outside mathematics, including general relativity.
In Group: Polynomials and Elementary Algebra, Lead section
formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric
View the SourceWikipedia: Isaac Newton
WikipediaPrincipal Result Of: Isaac Newton, Article textQuote, Principal Result Of: Isaac Newton, Article text
" He discovered Newton's identities (probably without knowing of earlier work by Albert Girard in 1629), Newton's method, the Newton polygon, and classified cubic plane curves (polynomials of degree three in two variables).
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