Reduction theorems for Noether’s problem

Let K be any field, G be a finite group. Let G act on the rational function field K(x(g) : g ∈ G) by K-automorphisms and h · x(g) = x(hg). Denote by K(G) = K(x(g) : g ∈ G) the fixed field. Noether’s problem asks whether K(G) is rational (= purely transcendental) over K. We will give several reduction theorems for solving Noether’s problem. For example, let G̃ = G × H be a direct product of finite groups. Theorem. Assume that either (1) H is an abelian group of exponent e and K contains a primitive e-th root of unity, or (2) K is a field with charK = p > 0 and H is a p-group. Then K(G̃) is rational over K(G). In particular, if K(G) is rational (resp. retract rational) over K, so is K(G̃) over K. 2000 Mathematics Subject Classification. Primary 12F12, 12F20, 13A50, 11R32, 14E08.