Noether's problem for S₄ and S₅

الأصل · EN

Let k be a field, G be a finite group and k(xg:g∈ G) be the rational function field over k, on which G acts by 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, i.e.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 Q(S₄) and Q(S₅) are not Q-rational. We will prove that, if k is a field such that char k ≠ 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.