On the question of the structure of the subfield of invariants of a cyclic group of automorphisms of the field $\mathbf Q(x_1,\dots,x_n)$

The subfield $L$ of the field $K=\mathbf Q(x_1,\dots,x_n)$ consisting of invariant functions relative to a cyclic permutation of the indeterminates is interpreted as the field of rational functions on a certain torus defined over $\mathbf Q$. On this basis, a necessary condition is derived for pure transcendence of $L$ over $\mathbf Q$ from which are obtained a number of counterexamples. A list is also given of fields $L$ which are purely transcendental over $\mathbf Q$.