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 $t\in (\bm{Z}/m\bm{Z})^\times$ is of order $n$. Define a group $G_{m,n}:=\langle\sigma,\tau:\sigma^m=\tau^n=1,\tau^{-1}\sigma\tau=\sigma^t\rangle$ $\simeq C_m \rtimes C_n$. 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 $\bm{C}(G_{p,n})$ is rational over $\bm{C} \}$ is of positive Dirichlet density; in particular, $S$ is an infinite set.