TITLE: How the ring of algebraic numbers looks like?
QUESTION [6 upvotes]: Suppose I have an algebraic number field $K = \mathbb Q(\alpha)$ for some $\alpha \in O_K$, ring of algebraic integers. Is there a criterion that tells us when $O_K =\mathbb Z[\alpha]$ by any chance?

For example, if $\alpha^3 - \alpha - 1 = 0$, then $O_K =\mathbb Z[\alpha]$. I was wondering if there is an easy way to see this.
Thanks!

REPLY [6 votes]: Such a basis for the ring of integers is called a power basis, and the corresponding field is said to be monogenic. See http://en.wikipedia.org/wiki/Monogenic_field.
Besides the cases mentioned in the wikipedia article, this is generally not an easy question to answer. See this MO question: https://mathoverflow.net/questions/21267/which-number-fields-are-monogenic-and-related-questions