Noether's Problem on Semidirect Product Groups

TL;DR

This paper investigates conditions under which the fixed field of a semidirect product group action on a rational function field is itself rational, providing new criteria especially for groups formed by cyclic groups of prime order.

Contribution

It introduces new criteria for the rationality of fixed fields for semidirect product groups, extending previous results and including specific conditions involving algebraic integers and norms.

Findings

Established new criteria for rationality of fixed fields for certain semidirect product groups.

Proved that if a prime p and q satisfy a norm condition in cyclotomic integers, then the fixed field is rational.

Extended known results to broader classes of groups beyond cyclic and abelian cases.

Abstract

Let $K$ be a field, $G$ a finite group. Let $G$ act on the function field $L = K(x_{\sigma} : \sigma \in G)$ by $\tau \cdot x_{\sigma} = x_{\tau\sigma}$ for any $\sigma, \tau \in G$. Denote the fixed field of the action by $K(G) = L^{G} = \left\{ \frac{f}{g} \in L : \sigma(\frac{f}{g}) = \frac{f}{g}, \forall \sigma \in G \right\}$. Noether's problem asks whether $K(G)$ is rational (purely transcendental) over $K$. It is known that if $G = C_m \rtimes C_n$ is a semidirect product of cyclic groups $C_m$ and $C_n$ with $\mathbb{Z}[\zeta_n]$ a unique factorization domain, and $K$ contains an $e$th 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 $K(C_m \rtimes C_n)$ is rational over $K$. In particular, if $p, q$ are prime numbers and there exists $x \in \mathbb{Z}[\zeta_q]$ such that the norm $N_{\mathbb{Q}(\zeta_q)/\mathbb{Q}}(x) = p$, then $\mathbb{C}(C_{p} \rtimes C_{q})$ is rational over $\mathbb{C}$.