Noether's problem for some semidirect products.

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 (h g) 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 / m Z) × is of order n. Define a group G m , n : = 〈 σ , τ : σ m = τ n = 1 , τ − 1 σ τ = σ t 〉 ≃ C m ⋊ C n. Assume furthermore that (i) m is an odd integer, and (ii) for any e | n , the ideal 〈 ζ e − t , m 〉 in Z [ ζ e ] is a principal ideal (where ζ e is a primitive e -th root of unity). Theorem. If k is a field with ζ m , ζ n ∈ k , then k (G m , n) is rational over k. Consequently, it may be shown that, for any positive integer n , the set S : = { p : p is a prime number such that C (G p , n) is rational over C } is of positive Dirichlet density; in particular, S is an infinite set. [ABSTRACT FROM AUTHOR]