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

TL;DR

This paper investigates the rationality properties of fixed fields under cyclic group actions on rational function fields, establishing non-stable rationality for certain primes over algebraic number fields.

Contribution

It extends Noether's problem by identifying conditions under which the fixed field is not stably rational, focusing on primes outside specific sets related to class number and ramification.

Findings

For primes p not in P_0 or P_k, the fixed field is not stably rational.

The result applies to algebraic number fields and primes with specific class number and ramification properties.

Provides new insights into the structure of invariant fields under cyclic group actions.

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 $k\}$.