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 $G=C_m⋊C_n$ 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 $K(C_m⋊C_n)$ 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 $\mathbb{C}(C_p⋊C_q)$ is rational over $\mathbb{C}$.

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• Noether's Problem for Some Semidirect Products