Primitive elements of number fields which span rings of integers My question is the following: given a number field $K$, does there exist a primitive element $\alpha$ of $K$ over $\mathbb{Q}$ such that the ring $\mathcal{O}_K$ of integers of $K$ is isomorphic to $\mathbb{Z}[\alpha]$?
For example, we have that the ring of integers of $\mathbb{Q}(\sqrt{d})$ (where $d$ is a square-free integer) the primitive element which satisfies the condition given above is $(1+\sqrt{d})/2$ or $\sqrt{d}$ whether $d$ is or not congruent to 1 modulo 4.

Dedekind showed that there is not always a primitive element/power basis for the ring of integers of a N.F.
His counter-example was the cubic field $\mathbb Q(\theta)$ where $\theta^3 + \theta^2 - 2\theta + 8$.
It is mentioned as one of the examples on wikipedia and there is a writeup about this in the note by B. Conrad:


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*https://en.wikipedia.org/wiki/Discriminant_of_an_algebraic_number_field

*http://math.stanford.edu/~conrad/154Page/handouts/nonprim.pdf
