Geodesic
On the question of the structure of the subfield of invariants of a cyclic group of automorphisms of the field $\mathbf Q(x_1,\dots,x_n)$
Izvestiya. Mathematics , Tome 4 (1970) no. 2, pp. 371-380

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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 $\mathbf Q$. 
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     title = {On the question of the structure of the subfield of invariants of a cyclic group of automorphisms of the field $\mathbf Q(x_1,\dots,x_n)$},
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V. E. Voskresenskii. On the question of the structure of the subfield of invariants of a cyclic group of automorphisms of the field $\mathbf Q(x_1,\dots,x_n)$. Izvestiya. Mathematics , Tome 4 (1970) no. 2, pp. 371-380. http://geodesic.mathdoc.fr/item/IM2_1970_4_2_a5/