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

TL;DR

This paper proves that for certain solvable subgroups of the symmetric group S_{14} over a field of characteristic 7, the fixed field of the group action on rational functions is rational, extending understanding of Noether's problem.

Contribution

It establishes the rationality of fixed fields for solvable transitive subgroups of S_{14} in characteristic 7, applying the Kuniyoshi-Gaschütz Theorem.

Findings

Fixed fields are rational for solvable transitive subgroups of S_{14} in characteristic 7.

Utilizes the Kuniyoshi-Gaschütz Theorem in the proof.

Advances understanding of Noether's problem in modular cases.

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 $\sigma\cdot x_{i}:=x_{\sigma\cdot i}$ for any $\sigma\in G$ 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-Gasch\"utz Theorem or some ideas in its proof.