On Noether's rationality problem for cyclic groups over $\mathbb{Q}$

Abstract

Let $p$ be a prime number. Let $C_p$, the cyclic group of order $p$, permute transitively a set of indeterminates $\{x_1,\dots ,x_p\}$. We prove that the invariant field $\mathbb{Q}(x_1,\dots ,x_p{)}^{C_p}$ is rational over $\mathbb{Q}$ if and only if the $(p-1)$-th cyclotomic field $\mathbb{Q}({\zeta }_{p-1})$ has class number one.

arXiv comment3 pages