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

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.