I have found that a lot of research has been done in rationality problem for algebraic tori. (For example, <https://arxiv.org/abs/1210.4525>). So I got to wonder what historical context or elementary motivation the problem has.



Historical context of rationality problem for algebraic torus
[Algebraic tori – thirty years after](https://arxiv.org/abs/0712.4061) gives some historical context:



> 
> The rationality problem goes back to the study of [Pythagorean triples](https://en.wikipedia.org/wiki/Pythagorean_triple) : Describe the set of solutions of a given system of
> polynomial equations by rational functions in a certain number of
> parameters. To establish rationality, one usually has to exhibit some
> explicit parameterization such as that obtained by stereographic
> projection in the case of Pythagoreas triples.
> 
> 
> In the converse "nonrationality" problem one wants to establish the
> non-existence of such a parameterization: here one seeks a birational
> invariant allowing one to detect nonrationality by comparing its value
> for the object under consideration with some “standard” one known to
> be zero; if the computation gives a nonzero value, we are done.
> Evidently, to be useful, such an invariant must be (relatively) easily
> computable.
> 
> 
> Algebraic tori are among the simplest algebraic groups, so much of the
> search for birotational invariants has focused on that topic.
> Pioneering results by Voskresenskii are reviewed in this paper.
> 
> 
> 


