Math 2.0

statement theorem

Fundamental theorem of symmetric polynomials

Statement

Let RR be a commutative ring and let f∈R[x1,…,xn]f \in R[x_1, \dots, x_n] be symmetric. Then there is a unique polynomial g∈R[y1,…,yn]g \in R[y_1, \dots, y_n] with f=g(e1,…,en),f = g(e_1, \dots, e_n), where e1,…,ene_1, \dots, e_n are the elementary symmetric polynomials.

Remarks

Equivalently, the map R[y1,…,yn]→R[x1,…,xn]R[y_1, \dots, y_n] \to R[x_1, \dots, x_n], yk↦eky_k \mapsto e_k, is injective, and its image is the ring of symmetric polynomials. One proof is by induction on the leading monomial in the lexicographic order. Each step subtracts a product of the eke_k with the same leading monomial. Example: x12+x22=e12−2e2x_1^2 + x_2^2 = e_1^2 - 2 e_2. So a symmetric expression in the roots of a polynomial over a field KK is an element of KK.

Explanations

L1 human

Every symmetric polynomial can be written in exactly one way using the elementary symmetric polynomials. The coefficients stay in the same ring.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionPolynomials in several variables, definition, no evidence levelPolynomials in severa…definitionSymmetric polynomial, definition, no evidence levelSymmetric polynomialdefinitionElementary symmetric polynomials, definition, no evidence levelElementary symmetric …definitionFundamental theorem of symmetric polynomials, statement, no evidence levelFundamental theorem o…statement
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  1. Group concept
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  8. Fundamental theorem of symmetric polynomials statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial, Fundamental theorem of symmetric polynomials

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