Let $p,q$ be odd primes. Consider the polynomial ring $\mathbb C[x_0,...,x_{q-1}]$. For $m=0,1,...,p-1$, let

$$\sigma_m=\sum_{0\le j_1\le p;...;0\le j_k\le p;0\le i_1\le q-1;...; 0\le i_j\le q-1; j_1+...+j_k=p; i_1j_1+...+i_kj_k\equiv m (\mod p)} \dfrac {p!}{j_1!...j_k!} x_{i_1}^{j_1}...x_{i_k}^{j_k}$$.

Notice that $\sigma_0+\sigma_1+...+\sigma_{p-1}=(x_1+...+x_{q-1})^p$ .

Let $K$ be the fraction field of $\mathbb C[\sigma_0,...,\sigma_{p-1}]$.

For which $p,q$, is it true that $\mathbb C(x_0,...,x_{q-1})$ is a finite Galois extension of $K$ ?

Let $p,q$ be odd primes. Consider the polynomial ring $\mathbb C[x_0,...,x_{q-1}]$. For $m=0,1,...,p-1$, let

$$\sigma_m=\sum_{0\le j_0\le p;...;0\le j_{q-1}\le p; j_1+...+j_{q-1}=p; 1.j_1+...+(q-1)j_{q-1}\equiv m (\mod p)} \dfrac {p!}{j_0!...j_{q-1}!} x_{0}^{j_0}...x_{q-1}^{j_{q-1}}$$.

Notice that $\sigma_0+\sigma_1+...+\sigma_{p-1}=(x_1+...+x_{q-1})^p$ .

Let $K$ be the fraction field of $\mathbb C[\sigma_0,...,\sigma_{p-1}]$.

For which $p,q$, is it true that $\mathbb C(x_0,...,x_{q-1})$ is a finite Galois extension of $K$ ?