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Let $p$ be a prime number. Let $C_p$, the cyclic group of order $p$, permutetransitively a set of indeterminates $\{ x_1,\ldots ,x_p \}$. We prove that theinvariant 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.

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