Noether's Problem on Semidirect Product Groups

Abstract

Let $K$ be a field, $G$ a finite group. Let $G$ act on the function field $L = K(x_{\sigma} : \sigma \in G)$ by $\tau \cdot x_{\sigma} = x_{\tau\sigma}$ for any $\sigma, \tau \in G$. Denote the fixed field of the action by $K(G) = L^{G} = \left\{ \frac{f}{g} \in L : \sigma(\frac{f}{g}) = \frac{f}{g}, \forall \sigma \in G \right\}$. Noether's problem asks whether $K(G)$ is rational (purely transcendental) over $K$. It is known that if $G = C_m \rtimes C_n$ is a semidirect product of cyclic groups $C_m$ and $C_n$ with $\mathbb{Z}[\zeta_n]$ a unique factorization domain, and $K$ contains an $e$th primitive root of unity, where $e$ is the exponent of $G$, then $K(G)$ is rational over $K$. In this paper, we give another criteria to determine whether $K(C_m \rtimes C_n)$ is rational over $K$. In particular, if $p, q$ are prime numbers and there exists $x \in \mathbb{Z}[\zeta_q]$ such that the norm $N_{\mathbb{Q}(\zeta_q)/\mathbb{Q}}(x) = p$, then $\mathbb{C}(C_{p} \rtimes C_{q})$ is rational over $\mathbb{C}$.