I have been studying towers of quadratic extensions to $\mathbb Q$ and have noticed the following:
$\mathbb Q[\sqrt 2]$ and $\mathbb Q[\sqrt 2][\sqrt 3]$ are unique factorization domains(UFDs), but $\mathbb Q[\sqrt 2][\sqrt 3][\sqrt 5]$ is not because
$$ 3 =\sqrt 3^2 = (\sqrt 5 + \sqrt 2)(\sqrt 5 - \sqrt 2) $$
I have not found much literature discussing towers of quadratic extensions and their relationship to various subclasses of integral domains. What are the conditions (if any) that towers of quadratic extensions to $\mathbb Q$ are UFDs, GCD domains, integrally closed domains, or simply integral domains? Is there a way to easily determine this? I'm also unclear on how the process of principalization transforms an integral domain into a UFD. The linked article on principalization uses very dense notation that I have difficulty following and I would like an illustrating example.