Fundamental theorem of symmetric polynomials - Imported Knowlpedia Content

Let $R$ be a commutative ring and consider the polynomial ring $R[x_1,\dots,x_n]$ .

A polynomial $f(x_1,\dots,x_n)\in R[x_1,\dots,x_n]$ is symmetric if

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

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 $f\in R[x_1,\dots,x_n]$ , there exists a unique polynomial $F\in R[t_1,\dots,t_n]$ such that

f(x_1,\dots,x_n)=F\big(e_1(x_1,\dots,x_n),\dots,e_n(x_1,\dots,x_n)\big).

Equivalent characterizations

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

R[x_1,\dots,x_n]^{\text{sym}} \cong R[e_1,\dots,e_n].

Remarks

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

Examples

For $n=2$ , with $e_1=x_1+x_2$ and $e_2=x_1x_2$ , $x_1^2+x_2^2 = (x_1+x_2)^2 - 2x_1x_2 = e_1^2 - 2e_2.$

For $n=2$ , $x_1^3+x_2^3 = (x_1+x_2)^3 - 3(x_1+x_2)(x_1x_2) = e_1^3 - 3e_1e_2.$

For $n=3$ , with $e_1=x_1+x_2+x_3$ and $e_2=x_1x_2+x_1x_3+x_2x_3$ , $x_1^2+x_2^2+x_3^2 = e_1^2 - 2e_2.$ (Note that $e_3=x_1x_2x_3$ is not needed in this particular expression, but it appears in many others.)