A note on Plans's paper of Noether's problem

Abstract

Let $p$ be a prime number and $\zeta_p$ be a primitive $p$-th root of unity in $\bm{C}$. Let $k$ be a field and $k(x_0,\ldots,x_{p-1})$ be the rational function field of $p$ variables over $k$. 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 $P_0=\{p\in P:\bm{Q}(\zeta_p)$ is of class number one$\}$. Theorem. If $k$ is an algebraic number field and $p\in P\backslash (P_0\cup P_k)$, then $k(x_0,\ldots,x_{p-1})^G$ is not stably rational over $k$ where $P_k=\{p\in P: p$ is ramified in $k\}$.