Abstract. Let K be any field and G be a finite group. Let G act on the rational function fields K(xg: g ∈ G) by K-automorphisms defined by g · xh = xgh for any g, h ∈ G. Denote by K(G) the fixed field K(xg: g ∈ G) G. Noether’s problem asks whether K(G) is rational (=purely transcendental) over K. We will prove that, if K is any field, p an odd prime number, and G is a non-abelian group of exponent p with |G | = p 3 or p 4 satisfying [K(ζp) : K] ≤ 2, then K(G) is rational over K. A notion of retract rationality is introduced by Saltman in case K(G) is not rational. We will also show that K(G) is retract rational if G belongs to a much larger class of p-groups. In particular, generic G-polynomials of G-Galois extensions exist for these groups
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