Noether's problem on semidirect product groups

Abstract

Let K be a field, G a finite group. Let G act on the function field L=K(xσ:σ∈G) by τ⋅xσ=xτσ for any σ,τ∈G. Denote the fixed field of the action by K(G)=LG={fg∈L:σ(fg)=fg,∀σ∈G}. Noether's problem asks whether K(G) is rational (purely transcendental) over K.It is known that if G=Cm⋊Cn is a semidirect product of cyclic groups Cm and Cn with Z[ζn] a unique factorization domain, and K contains an eth primitive root of unity, where e is the exponent of G, then K(G) is rational over K.In this paper, we give another criteria to determine whether K(Cm⋊Cn) is rational over K. In particular, if p,q are prime numbers and there exists x∈Z[ζq] such that the norm NQ(ζq)/Q(x)=p, then C(Cp⋊Cq) is rational over C.