Given a field $K$ (say, of characteristic zero) and $K[x_1,\dots,x_n]$ the ring of polynomials in $n$-variables, consider the action of the cyclic group $C_n=<\sigma>$ with $n$ elements by $\sigma(x_i)=x_{i+1}$ if $i<n$, $\sigma(x_n)=x_1$. The polynomials $p(x_1,\dots,x_n)$ such that $\sigma(p)=p$ are called cyclic polynomials.
Question: Can we describe the ring $CP_n$ of cyclic polynomials, giving generators and relations?
If $n=2$, then cyclic is the same than symmetric, hence the two elementary symmetric polynomials $s_1$ and $s_2$ generate and there are no relations. So $CP_2=K[s_1,s_2]$.
If $n=3$, then cyclic is the same than alternating, and it is known in this case that the elementary symmetric polynomials $s_1,s_2,s_3$ plus the polynomial $d:=(x_1-x_2)(x_2-x_3)(x_1-x_3)$ generate. But $d^2=\Delta$ is the discriminant, which can be express as a polynomial in $s_1,s_2,s_3$. Then $$CP_3=K[s_1,s_2,s_3,d]/(d^2-\Delta).$$
Can we give a description as above in the cases $n=4$ and $n=5$?