Vol. 258, No. 2, 2012

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Noether’s problem for Ŝ4 and Ŝ5

Ming-chang Kang and Jian Zhou

Vol. 258 (2012), No. 2, 349–368
Abstract

Let k be a field, let G be a finite group and let k(xg : g G) be the rational function field over k, on which G acts by the k-automorphisms defined by h xg = xhg for any g,h G. Noether’s problem asks whether the fixed subfield k(G) := k(xg : g G)G is k-rational, that is, purely transcendental over k. If Sn is the double cover of the symmetric group Sn, in which the liftings of transpositions and products of disjoint transpositions are of order 4, Serre shows that (S4) and (S5) are not -rational. We will prove that if k is a field such that chark2,3, 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
Authors
Ming-chang Kang
Department of Mathematics and
Taida Institute of Mathematical Sciences
National Taiwan University
Taipei 106
Taiwan
Jian Zhou
School of Mathematical Sciences
Peking University
Beijing, 100871
China