Noether's Problem for Abelian Extensions of Cyclic p-groups of Nilpotency Class 2

Abstract

Let $$K$$ be a field and $$G$$ be a finite group. Let $$G$$ act on the rational function field $$K(x(g):g\in G)$$ by $$K$$-automorphisms defined by $$g\cdot x(h)=x(gh)$$ for any $$g,h\in G$$. Denote by $$K(G)$$ the fixed field $$K(x(g):g\in G)^G$$. Noether's problem then asks whether $$K(G)$$ is rational over $$K$$. Let $$p$$ be prime and let $$G$$ be a $$p$$-group of exponent $$p^e$$. Assume also that (i) $$K = p>0$$, or (ii) $$K \ne p$$ and $$K$$ contains a primitive $$p^e$$-th root of unity. In this paper we prove that $$K(G)$$ is rational over $$K$$ if $$G$$ is any finite $$p$$-group of nilpotency class $$2$$ which is an abelian extension of a cyclic group.