The symmetric polynomials as the invariant ring of the symmetric group, seen through Noether's finiteness theorem

The permutation action is by $k$-algebra automorphisms. For $\sigma \in {\mathrm{Sym}}_n$ the map ${\mathrm{\Phi }}_{\sigma }$ sending $x_j$ to $x_{\sigma (j)}$ and fixing $k$ is a unital ring homomorphism, obtained by iterating the universal property of a polynomial ring along the tower defining $C$; composing, ${\mathrm{\Phi }}_{\sigma }{\mathrm{\Phi }}_{\tau }$ sends $x_j$ to $x_{\sigma (\tau (j))}$, so ${\mathrm{\Phi }}_{\sigma }{\mathrm{\Phi }}_{\tau }={\mathrm{\Phi }}_{\sigma \tau }$ and ${\mathrm{\Phi }}_{\sigma }$ is bijective with inverse ${\mathrm{\Phi }}_{{\sigma }^{-1}}$. Each ${\mathrm{\Phi }}_{\sigma }$ fixes the constants, so the action is by $k$-algebra automorphisms, and the invariant subring $C^{{\mathrm{Sym}}_n}$ is by definition the set of $f$ with $\sigma \cdot f=f$ for all $\sigma$, which is the ring of symmetric polynomials.

Noether's theorem applies. The ring $k$ is Noetherian, $C=k[x_1,\dots ,x_n]$ is of finite type over $k$ with the indeterminates as generators, $k$ is a subring of $C$, and ${\mathrm{Sym}}_n$ is a finite group acting on $C$ by $k$-algebra automorphisms. So $C^{{\mathrm{Sym}}_n}$ is of finite type over $k$: some finite list of symmetric polynomials generates it as a $k$-algebra. The theorem exhibits no such list.

The classical theorem gives strictly more, and the two agree where they overlap. Substitution $T_j\mapsto e_j$ is a $k$-algebra isomorphism onto $C^{{\mathrm{Sym}}_n}$, so $C^{{\mathrm{Sym}}_n}=k[e_1,\dots ,e_n]$, which in particular is a finite generating list and so reproves finite type. The extra content is twofold: the generators are named, and the isomorphism is injective, so the expression of a symmetric polynomial as a polynomial in $e_1,\dots ,e_n$ is unique. Nothing in Noether's theorem gives either.

The orbit polynomial at $x_1$ is a power of the Vieta product. By the action formula $\sigma \cdot x_1=x_{\sigma (1)}$, so $P_{x_1}(T)={\prod }_{\sigma }(T-x_{\sigma (1)})$, in which the factor $T-x_i$ occurs once for each $\sigma$ with $\sigma (1)=i$. For indices $i\mathrm{\ne }{i}^{\mathrm{\prime }}$, composing with the transposition exchanging $i$ and $i^{\mathrm{\prime }}$ is a bijection between the permutations with $\sigma (1)=i$ and those with $\sigma (1)=i^{\mathrm{\prime }}$, so all these counts are equal, say to $q$; summing over $i$ gives $nq=\mathrm{\mid }{\mathrm{Sym}}_n\mathrm{\mid }$. Hence $P_{x_1}(T)=({\prod }_{i=1}^n(T-x_i){)}^q$, and the coefficients of ${\prod }_{i=1}^n(T-x_i)$ are the elementary symmetric polynomials in $x_1,\dots ,x_n$ up to sign, the coefficient of $T^{n-k}$ being $(-1{)}^k{e}_k$. Every coefficient of $P_{x_1}$ is therefore a polynomial in $e_1,\dots ,e_n$, consistent with the general statement that the coefficients lie in the invariant subring.