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 \(p\in P\backslash (P_0\cup P_k)\).