# Review on rationality problems of algebraic k-tori

## TL;DR

This paper reviews the rationality issues of algebraic k-tori, linking them to invariant fields and Noether's Problem, and discusses methods to determine their rationality.

## Contribution

It provides a comprehensive overview of rationality problems for algebraic k-tori, including character group analysis and numerical approaches for rationality determination.

## Key findings

- Rationality of k-tori is equivalent to the rationality of their invariant function fields.
- Character groups are useful tools in studying k-tori rationality.
- Numerical methods can assist in determining the rationality of algebraic k-tori.

## Abstract

Rationality problems of algebraic k-tori are closely related to rationality problems of the invariant field, also known as Noether's Problem. We describe how a function field of algebraic k-tori can be identified as an invariant field under a group action and that a k-tori is rational if and only if its function field is rational over k. We also introduce character group of k-tori and numerical approach to determine rationality of k-tori.