Noether's problem for some subgroups of $S_{14}$: the modular case

Abstract

Let $G$ be a subgroup of $S_{n}$, the symmetric group of degree $n$. For any field $k$, $G$ acts naturally on the rational function field $k(x_{1},\cdots,x_{n})$ via $k$-automorphisms defined by $σ\cdot x_{i}:=x_{σ\cdot i}$ for any $σ\in G$ and $1\leq i\leq n$. In this article, we will show that if $G$ is a solvable transitive subgroup of $S_{14}$ and $\text{char}(k)=7$, then the fixed subfield $k(x_{1},\cdots,x_{14})^{G}$ is rational (i.e., purely transcendental) over $k$. In proving the above theorem, we rely on the Kuniyoshi-Gaschütz Theorem or some ideas in its proof.