TITLE: Is the ring $\mathbb{Z}[1,2^{1/n}]$ a UFD?
QUESTION [5 upvotes]: I'm a bit rusty on my ring theory and I'm working on a pet project doing some numerical work with primes in rings. I couldn't find answers to this online, so I just want to know if  $\mathbb{Z}[1,2^{1/n}]$ is a Unique Factorization Domain (so I can talk about primes sensibly). I have no idea how to go about this personally.
For my purposes it need not be true for all $n$, it would be enough that there is a closed form giving $n$ arbitrarily large for which it works, e.g. perhaps $n=2^k$? (That said, something a bit slower than an exponential would be appreciated for numerical work).

REPLY [4 votes]: Some people believe that the answer is yes. Using pari you can compute the class groups almost up to $n=40$; all of these fields seem to have class number 
$1$. For some subfamilies (i.e. $n = 2^k$) you may prove that the class number is odd. Other than that, nothing is known. 
I do not think that a general proof is possible without having a complete system of independent units. I did not succeed in finding one.