Noether's Problem for Some Semidirect Products

2017/03/03 by Kang, Ming-chang, Zhou, Jian
#11R29 #12F10 #13A50 #14E08 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1703.01010

Abstract

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∈ (\bmZ/m\bmZ)^× is of order n. Define a group G_m,n:=⟨σ,τ:σ^m=τⁿ=1,τ^-1στ=σ^t⟩ ≃ C_m \rtimes C_n. 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 \bmC(G_p,n) is rational over \bmC \ is of positive Dirichlet density; in particular, S is an infinite set.

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