Noether’s problem for Ŝ₄ and Ŝ₅

Ming-chang Kang and Jian Zhou

Vol. 258 (2012), No. 2, 349–368

DOI: 10.2140/pjm.2012.258.349

Abstract

Let k be a field, let G be a finite group and let k(x_g : g ∈ G) be the rational function field over k, on which G acts by the k-automorphisms defined by h ⋅ x_g = x_hg for any g,h ∈ G. Noether’s problem asks whether the fixed subfield k(G) := k(x_g : g ∈ G)^G is k-rational, that is, purely transcendental over k. If S_n is the double cover of the symmetric group S_n, in which the liftings of transpositions and products of disjoint transpositions are of order 4, Serre shows that ℚ(S₄) and ℚ(S₅) are not ℚ-rational. We will prove that if k is a field such that chark≠2,3, and k(ζ₈) is a cyclic extension of k, then k(S₄) is k-rational. If it is assumed furthermore that chark = 0, then k(S₅) is also k-rational.

Keywords

Noether’s problem, rationality problem, binary octahedral groups

Mathematical Subject Classification 2010

Primary: 14E08, 14M20

Secondary: 12F12, 13A50

Milestones

Received: 11 October 2011

Revised: 26 December 2011

Accepted: 5 January 2012

Published: 1 August 2012