Historical context of rationality problem for algebraic torus 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.
 A: Algebraic tori – thirty years after gives some historical context:

The rationality problem goes back to the study of Pythagorean triples : 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.

