Noether's Problem for Some Semidirect Products

الأصل · EN

Let k be a field, G be a finite group, k(x(g):g∈ G) be the rational function field with the variables x(g) where g∈ G. The group G acts on k(x(g):g∈ G) by k-automorphisms where h· x(g)=x(hg) for all h,g∈ G. Let k(G) be the fixed field defined by k(G):=k(x(g):g∈ G)ᵍ={f∈ k(x(g):g∈ G): h· f=f for all h∈ G}. Noether's problem asks whether the fixed field k(G) is rational (= purely transcendental) over k. Let m and n be positive integers and assume that there is an integer t such that t∈ (Z/mZ)× is of order n. Define a group Gₘ,ₙ:=σ,τ:σᵐ=τⁿ=1,τ⁻¹στ=σᵗ ≃ Cₘ Cₙ. We will find a sufficient condition to guarantee that k(G) is rational over k. As a result, it is shown that, for any positive integer n, the set S:={p: p is a prime number such that C(Gₚ,ₙ) is rational over C } is of positive Dirichlet density; in particular, S is an infinite set.