Start with the cyclic group $G:=\mathbb{Z}/p$ of prime order $p$ and and an integer lattice $P:=\mathbb{Z}^p$. Let $G$ act on $P$ by cyclic permutation of coordinates. There is an induced action on the group ring $\mathbb{Z}[P]$ of the lattice $P$ which is isomorphic to the ring $\mathbb{Z}[\mathbf{x^{\pm 1}}]$ of Laurent polynomials in $p$ variables $\mathbf{x}=(x_1,\dots,x_p)$ over $\mathbb{Z}$.

How do I find a (hopefully) finite/minimal set of invariants which generates the ring of $G$-invariants $\mathbb{Z}[\mathbf{x^{\pm 1}}]^G$?

Since $\mathbb{Z}[\mathbf{x^{\pm 1}}]^G=\mathbb{Z}[\mathbf{x}]^G[\sigma_p^{-1}]$ where $\sigma_p=x_1\dots x_p$, the question about Invariant Polynomials under a Group Action (hidden GIT) is of some help but not much is said there about what happens over $\mathbb{Z}$.

From the Lie theory point of view this is related to the weight lattice of the root system $A_{p-1}$ where $G$ is a subgroup of the Weyl group $S_p$ and the fundamental domain I'm considering is some sort of "larger Weyl chamber".