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Fundamental Theorem of Symmetric Polynomials

Every symmetric polynomial is a polynomial in the elementary symmetric polynomials.

Fundamental Theorem of Symmetric Polynomials

Theorem.
Let $R$ be a commutative ring and consider $R[x_1,\dots,x_n]$ (a polynomial ring ). A polynomial $f\in R[x_1,\dots,x_n]$ is symmetric if it is invariant under every permutation of the variables.

Then every symmetric polynomial $f$ can be written uniquely as

f = F(e_1,\dots,e_n)

for some $F\in R[t_1,\dots,t_n]$ , where $e_k$ is the $k$ -th elementary symmetric polynomial:

e_1=\sum_i x_i,\quad e_2=\sum_{i<j}x_ix_j,\ \dots,\ e_n=x_1\cdots x_n.

Examples.

For $n=2$ , $x_1^2+x_2^2 = (x_1+x_2)^2 - 2x_1x_2 = e_1^2-2e_2$ .
For $n=3$ , $x_1^2+x_2^2+x_3^2 = e_1^2 - 2e_2$ .
For $n=2$ , $x_1^3+x_2^3 = (x_1+x_2)^3 - 3(x_1+x_2)(x_1x_2)= e_1^3-3e_1e_2$ .