NOETHER'S PROBLEM FOR ABELIAN EXTENSIONS OF CYCLIC p-GROUPS

摘要

Let K be a field and G 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: g e G)~G. Noether's problem then asks whether K(G) is rational (i.e., purely transcendental) over K. The first main result of this article is that K(G) is rational over K for a certain class of p-groups having an abelian subgroup of index p. The second main result is that K(G) is rational over K for any group of order p~5 or p~6 (where p is an odd prime) having an abelian normal subgroup such that its quotient group is cyclic. (In both theorems we assume that if char K ≠ p then K contains a primitive p~e-th root of unity, where p~e is the exponent of G.)