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

TL;DR

This paper investigates the rationality of invariant fields under cyclic group actions over the rationals, establishing a precise criterion linked to cyclotomic field class numbers.

Contribution

It provides a complete characterization of when the invariant field under a cyclic group action over  is rational, based on cyclotomic field class number conditions.

Findings

Invariant field is rational iff ((-1)) has class number one

Connects rationality problem to class number of cyclotomic fields

Provides a criterion for cyclic group actions over 

Abstract

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