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 $k(x_g\mid g\in G)$ by $k$-automorphisms $h(x_g)=x_{hg}$ for any $g,h\in G$. Noether’s problem asks whether the invariant field $k(G)=k(x_g\mid g\in G{)}^G$ 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 $\mathbf{Q}(C_p)$ is rational over $\mathbf{Q}$. Only known 17 primes $p$ for which $\mathbf{Q}(C_p)$ is rational over $\mathbf{Q}$ are $p\le 43$ and $p=61,67,71$. We show that for primes $p<20000$, $\mathbf{Q}(C_p)$ is not (stably) rational over $\mathbf{Q}$ except for affirmative 17 primes and undetermined 46 primes. Under the GRH, the generalized Riemann hypothesis, we also confirm that $\mathbf{Q}(C_p)$ is not (stably) rational over $\mathbf{Q}$ for undetermined 28 primes $p$ out of 46.