On Noether's problem for cyclic groups of prime order

Abstract

Let $k$ be a field and $G$ be a finite group acting on the rational function field $k(x_g\,|\,g\in G)$ by $k$-automorphisms $h(x_g)=x_{hg}$ for any $g,h\in G$. Noether's problem asks whether the invariant field $k(G)=k(x_g\,|\,g\in G)^G$ is rational (i.e. purely transcendental) over $k$. In 1974, Lenstra gave a necessary and sufficient condition to this problem for abelian groups $G$. However, even for the cyclic group $C_p$ of prime order $p$, it is unknown whether there exist infinitely many primes $p$ such that $\mathbb{Q}(C_p)$ is rational over $\mathbb{Q}$. Only known $17$ primes $p$ for which $\mathbb{Q}(C_p)$ is rational over $\mathbb{Q}$ are $p\leq 43$ and $p=61,67,71$. We show that for primes $p< 20000$, $\mathbb{Q}(C_p)$ is not (stably) rational over $\mathbb{Q}$ except for affirmative $17$ primes and undetermined $46$ primes. Under the GRH, the generalized Riemann hypothesis, we also confirm that $\mathbb{Q}(C_p)$ is not (stably) rational over $\mathbb{Q}$ for undetermined $28$ primes $p$ out of $46$.