# Rationality of Three-Dimensional Quotients by Monomial Actions

## TL;DR

This paper proves that for certain finite 2-groups acting monomially on a three-variable rational function field over a field with specific properties, the fixed field remains rational, extending understanding of field invariants.

## Contribution

It establishes the rationality of fixed fields under monomial actions of finite 2-groups on three-variable rational function fields, under particular field conditions.

## Key findings

- Fixed field $K(x,y,z)^G$ is rational over $K$ for the specified group actions.
- The result applies to fields with characteristic not 2 and containing square roots of all elements.
- Provides applications demonstrating the theorem's utility.

## Abstract

Let $G$ be a finite 2-group and $K$ be a field satisfying that (i) $\fn{char}K\ne 2$, and (ii) $\sqrt{a}\in K$ for any $a\in K$. If $G$ acts on the rational function field $K(x,y,z)$ by monomial $K$-automorphisms, then the fixed field $K(x,y,z)^G$ is rational (= purely transcendental) over $K$. Applications of this theorem will be given.