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Fundamental theorem of symmetric polynomials

A symmetric polynomial can be expressed uniquely in terms of the elementary symmetric polynomials.
Fundamental theorem of symmetric polynomials

Let RR be a and consider the polynomial ring R[x1,…,xn]R[x_1,\dots,x_n].

A polynomial f(x1,…,xn)∈R[x1,…,xn]f(x_1,\dots,x_n)\in R[x_1,\dots,x_n] is symmetric if

f(xσ(1),…,xσ(n))=f(x1,…,xn)for every permutation σ of {1,…,n}. f(x_{\sigma(1)},\dots,x_{\sigma(n)}) = f(x_1,\dots,x_n) \quad\text{for every permutation }\sigma\text{ of }\{1,\dots,n\}.

Define the elementary symmetric polynomials

ek(x1,…,xn)=∑1≤i1<⋯<ik≤nxi1⋯xik(k=1,…,n). e_k(x_1,\dots,x_n)=\sum_{1\le i_1<\cdots<i_k\le n} x_{i_1}\cdots x_{i_k} \qquad (k=1,\dots,n).

Theorem (Fundamental theorem of symmetric polynomials). For every symmetric polynomial f∈R[x1,…,xn]f\in R[x_1,\dots,x_n], there exists a unique polynomial F∈R[t1,…,tn]F\in R[t_1,\dots,t_n] such that

f(x1,…,xn)=F(e1(x1,…,xn),…,en(x1,…,xn)). f(x_1,\dots,x_n)=F\big(e_1(x_1,\dots,x_n),\dots,e_n(x_1,\dots,x_n)\big).

Equivalently, the subring of symmetric polynomials is a polynomial ring:

R[x1,…,xn]sym≅R[e1,…,en]. R[x_1,\dots,x_n]^{\text{sym}} \cong R[e_1,\dots,e_n].

This theorem is a key bridge to : if f∈K[x]f\in K[x] splits over a as ∏i=1n(x−αi)\prod_{i=1}^n (x-\alpha_i), then the coefficients of ff are (up to signs) exactly the elementary symmetric polynomials in the roots αi\alpha_i.

Examples

  1. For n=2n=2, with e1=x1+x2e_1=x_1+x_2 and e2=x1x2e_2=x_1x_2,

    x12+x22=(x1+x2)2−2x1x2=e12−2e2. x_1^2+x_2^2 = (x_1+x_2)^2 - 2x_1x_2 = e_1^2 - 2e_2.

  2. For n=2n=2,

    x13+x23=(x1+x2)3−3(x1+x2)(x1x2)=e13−3e1e2. x_1^3+x_2^3 = (x_1+x_2)^3 - 3(x_1+x_2)(x_1x_2) = e_1^3 - 3e_1e_2.

  3. For n=3n=3, with e1=x1+x2+x3e_1=x_1+x_2+x_3 and e2=x1x2+x1x3+x2x3e_2=x_1x_2+x_1x_3+x_2x_3,

    x12+x22+x32=e12−2e2. x_1^2+x_2^2+x_3^2 = e_1^2 - 2e_2.

    (Note that e3=x1x2x3e_3=x_1x_2x_3 is not needed in this particular expression, but it appears in many others.)