Noether’s problem for cyclic groups of prime order

Abstract

Let k be a field and $k(x_0,\ldots ,x_{p-1})$ be the rational function field of p variables over k where p is a prime number. Suppose that $G=\langle \sigma \rangle \simeq C_p$ acts on $k(x_0,\ldots ,x_{p-1})$ by k-automorphisms defined as $\sigma :x_0\mapsto x_1\mapsto \cdots \mapsto x_{p-1}\mapsto x_0$. Denote by P the set of all prime numbers and define $P_0=\{p\in P:\mathbb {Q}(\zeta _{p-1})$ is of class number one$\}$ where $\zeta _n$ a primitive n-th root of unity in $\mathbb {C}$ for a positive integer n; $P_0$ is a finite set by Masley and Montgomery (J Reine Angew Math 286/287:248–256, 1976). Theorem. Let k be an algebraic number field and $P_k=\{p\in P: p$ is ramified in $k\}$. Then $k(x_0,\ldots ,x_{p-1})^G$ is not stably rational over k for all $p\in P\backslash (P_0\cup P_k)$.