What are symmetric polynomials? State their fundamental structure theorem.
Solution
A polynomial \[ f(x_1,\dots,x_n)\in R[x_1,\dots,x_n] \] is symmetric if \[ f(x_{\sigma(1)},\dots,x_{\sigma(n)})=f(x_1,\dots,x_n) \] for every permutation \(\sigma\in S_n\).
The basic examples are the elementary symmetric polynomials \[ e_k=\sum_{1\le i_1<\cdots<i_k\le n}x_{i_1}\cdots x_{i_k}, \qquad 1\le k\le n. \]
The fundamental theorem of symmetric polynomials says that every symmetric polynomial \(f\) can be written uniquely as a polynomial in \[ e_1,\dots,e_n. \] Equivalently, \[ R[x_1,\dots,x_n]^{S_n}=R[e_1,\dots,e_n]. \] Thus the ring of symmetric polynomials is itself a polynomial ring on the elementary symmetric functions.