Every finite extension $K$ of $\mathbb Q$ can be written as $\mathbb Q[\alpha]$. A very naive but important question is to ask if the ring of integers of $K$ is equal to $\mathbb Z[\alpha]$. When is this true?
Small known examples:
For quadratic field $\mathbb Q[\sqrt{n}]$, its ring of integer is either $\mathbb Z[\sqrt n]$ or $\mathbb Z[\frac12(1+\sqrt n)]$ when $n\equiv 2,3(\text{mod 4})$ or $n\equiv 1(\text{mod 4})$ respectively.
For cyclotomic field $\mathbb Q[\zeta_n]$, its ring of integers is $\mathbb Z[\zeta_n]$.
Do we have answers for other simple cases like $\mathbb Q[n^{1/3}]$? What can we say in general?