Is $\mathbb{Z} [(-2)^{\frac{1}{n}}]$ ever a UFD?

Question

Let $n>2$ be an integer.

Consider the integral domains $\mathbb{Z} [(-2)^{\frac{1}{n}}]$.

For what integer values of $n$ is $\mathbb{Z} [(-2)^{\frac{1}{n}}]$ a UFD ?

Are there infinitely many solutions ?

Same question for $\mathbb{Z} [2^{\frac{1}{n}}]$.