Noether’s problem for dihedral 2-groups II

Ming-chang Kang

Vol. 222 (2005), No. 2, 301–316

DOI: 10.2140/pjm.2005.222.301

Abstract

Let K be any field and G be a finite group. Let G act on the rational function field K(x_g : g ∈ G) by K-automorphisms defined by g ⋅x_h = x_gh for any g,h ∈ G. Denote by K(G) the fixed field K(x_g : g ∈ G)^G. Noether’s problem asks whether K(G) is rational (= purely transcendental) over K. A result of Serre shows that ℚ(G) is not rational when G is the generalized quaternion group of order 16. We shall prove that K(G) is rational over K if G is any nonabelian group of order 16 except when G is the generalized quaternion group of order 16. When G is the generalized quaternion group of order 16 and K(ζ₈) is a cyclic extension of K, then K(G) is also rational over K.

Keywords

rationality, Noether’s problem, generic Galois extensions, generic polynomials, groups of order 16

Mathematical Subject Classification 2000

Primary: 12F12, 13A50, 11R32, 14E08

Milestones

Published: 1 December 2005