# Noether's problem for dihedral 2-groups

### Huah Chu

National Taiwan University, Taipei, Taiwan### Ming-Chang Kang

National Taiwan University, Taipei, Taiwan### Shou-Jen Hu

Tamkang University, Taipei, Taiwan

## 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 $K(x_g: \, g \in G)^G$. Noethers problem asks whether $K(G)$ is rational (= purely transcendental) over $K$. We shall prove that $K(G)$ is rational over $K$ if $G$ is the dihedral group (resp. quasi-dihedral group, modular group) of order 16. Our result will imply the existence of the generic Galois extension and the existence of the generic polynomial of the corresponding group.

## Cite this article

Huah Chu, Ming-Chang Kang, Shou-Jen Hu, Noether's problem for dihedral 2-groups. Comment. Math. Helv. 79 (2004), no. 1, pp. 147–159

DOI 10.1007/S00014-003-0783-8