On Rationality of Fields of Invariants for Abelian Groups of Odd Order

In this paper, we investigate Noether’s problem concerning the rationality of the field of invariants k(G) for finite abelian groups G of odd order. We establish necessary and sufficient conditions for the rationality of the extension k(G) over an arbitrary field k of characteristic 0, providing a more flexible alternative to the classical requirement that all cyclic primary components be rational. Specifically, we present a criterion for elementary abelian q-groups Cqn in terms of the properties of certain norm maps (Theorem 4) and generalize these results to provide necessary and sufficient conditions for arbitrary abelian groups of odd order (Theorem 5). Furthermore, we provide a computational implementation of these criteria in PARI/GP and offer a concrete arithmetic classification of rational invariant fields over Q for groups of various odd orders.