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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?

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    The key word is "monogenic extension". – 2017-02-02
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    Thank you, I found a rather detailed question on MO: https://mathoverflow.net/questions/21267/which-number-fields-are-monogenic-and-related-questions – 2017-02-02

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