Action of dihedral groups

TL;DR

This paper surveys Noether's problem concerning the rationality of fixed fields under group actions, focusing on abelian and dihedral groups, and proves rationality for dihedral groups of order up to 10 over the rationals.

Contribution

It provides a survey of Noether's problem for specific groups and establishes rationality results for dihedral groups of small order over the rationals.

Findings

$ ext{Q}(D_n)$ is rational over $ ext{Q}$ for $n extless 11$

Survey of Noether's problem for abelian and dihedral groups

Extension of known rationality results for dihedral groups

Abstract

Let $K$ be any field and $G$ be a finite group. Let $G$ act on the rational function field $K(x_g: \ g \in G)$ by $K$-automorphisms defined by $g \cdot x_h=x_{gh}$ for any $g, \ h \in G$. Denote by $K(G)$ the fixed field $K(x_g: \ g \in G)^G$. Noether's problem asks whether $K(G)$ is rational (=purely transcendental) over $K$. We will give a brief survey of Noether's problem for abelian groups and dihedral groups, and will show that $\Bbb Q(D_n)$ is rational over $\Bbb Q$ for $n \le 10$.