A number field $K$ is said to be monogenic when $mathcal{O}_K=mathbb{Z}[alpha]$ for some $alphainmathcal{O}_K$. What is currently known about which $K$ are monogenic? Which are not? From Marcus’s *Number Fields*, I’m familiar with the proof that the cyclotomic fields are monogenic, and for example that $mathbb{Q}(sqrt{7},sqrt{10})$ is not monogenic (it is exercise 30 of chapter 2), but because Marcus eschews anything local, I haven’t seen any of the perhaps more natural proofs of these results.

If $K$ is monogenic, is there an effective method of determining those $alphainmathcal{O}_K$ for which $mathcal{O}_K=mathbb{Z}[alpha]$?

More generally, what is known about the minimal number of generators of $mathcal{O}_K$ as a $mathbb{Z}$-algebra? That is, can we determine, or at least put non-trivial bounds on, the minimal $m$ such that $mathcal{O}_K=mathbb{Z}[alpha_1,ldots,alpha_m]$ for some $alpha_iinmathcal{O}_K$? We know that any $mathcal{O}_K$ has an integral basis of $n=[K:mathbb{Q}]$ elements, so certainly $mleq n$ (I’m considering that trivial).