Noether's Problem for Some Semidirect Products

Abstract

Let $k$ be a field, $G$ be a finite group, $k(x(g):g\in G)$ be the rational function field with the variables $x(g)$ where $g\in G$. The group $G$ acts on $k(x(g):g\in G)$ by $k$-automorphisms where $h\cdot x(g)=x(hg)$ for all $h,g\in G$. Let $k(G)$ be the fixed field defined by $k(G):=k(x(g):g\in G{)}^G=\{f\in k(x(g):g\in G):h\cdot f=f$ for all $h\in G\}$. Noether's problem asks whether the fixed field $k(G)$ is rational (= purely transcendental) over $k$. Let $m$ and $n$ be positive integers and assume that there is an integer $t$ such that $t\in (\mathbf{Z}/m\mathbf{Z}{)}^{\times }$ is of order $n$. Define a group $G_{m,n}:=⟨\sigma ,\tau :{\sigma }^m={\tau }^n=1,{\tau }^{-1}\sigma \tau ={\sigma }^t⟩$ $\simeq C_m⋊C_n$. We will find a sufficient condition to guarantee that $k(G)$ is rational over $k$. As a result, it is shown that, for any positive integer $n$, the set $S:=\{p:p$ is a prime number such that $\mathbf{C}(G_{p,n})$ is rational over $\mathbf{C}\}$ is of positive Dirichlet density; in particular, $S$ is an infinite set.