Math 2.0

Combinatorics Theorem

Fundamental theorem of symmetric polynomials

Builds on Symmetric polynomialElementary 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.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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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
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Polynomial ring definition
  5. Polynomials in several variables definition
  6. Symmetric polynomial definition
  7. Elementary symmetric polynomials definition
  8. Fundamental theorem of symmetric polynomials statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial, Fundamental theorem of symmetric polynomials
About this page
Tag0GW1ZZH0R4
URIm2://0GW1ZZH0R4
Kindstatement, theorem
WritingW0 stub
MSC05E05
StatusThis copy of the site has no build data. Run m2 status 0GW1ZZH0R4.
UsesSymmetric polynomial, Elementary symmetric polynomials

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