JournalscmhVol. 79, No. 1pp. 147–159

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
Noether's problem for dihedral 2-groups cover

Abstract

Let KK be any field and GG be a finite group. Let GG act on the rational function field K(xg:gG)K(x_g: \, g \in G) by KK-automorphisms defined by gxh=xghg \cdot x_h= x _{gh} for any g,hGg, \, h \in G. Denote by K(G)K(G) the fixed field K(xg:gG)GK(x_g: \, g \in G)^G. Noethers problem asks whether K(G)K(G) is rational (= purely transcendental) over KK. We shall prove that K(G)K(G) is rational over KK if GG 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