TITLE: Which number fields are monogenic? and related questions
QUESTION [39 upvotes]: A number field $K$ is said to be monogenic when $\mathcal{O}_K=\mathbb{Z}[\alpha]$ for some $\alpha\in\mathcal{O}_K$. 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 $\alpha\in\mathcal{O}_K$ 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 $\alpha_i\in\mathcal{O}_K$? We know that any $\mathcal{O}_K$ has an integral basis of $n=[K:\mathbb{Q}]$ elements, so certainly $m\leq n$ (I'm considering that trivial).

REPLY [6 votes]: I recently ran across the following result in a paper of [Ash-Brakenhoff-Zarrabi][1] (Lemma 3.1), where it is called Dedekind's criterion:
Let $T(x)$ be a monic irreducible polynomial with root $\theta$ and let $K = \mathbb{Q}(\theta)$. Let $\mathcal{O}$ be the ring of integers in $K$ and let $p$ be a prime. We use overlines to denote reduction modulo $p$.
Factor $\bar{T}(x)$ in $\mathbb{F}_p[x]$ as $\prod \bar{t}_i(x)^{e_i}$, and choose lifts of the $\bar{t}_i$ to monic polynomials in $\mathbb{Z}[x]$. Define $g(x) = \prod t_i(x)$, $h(x)=\prod t_i(x)^{e_i-1}$ and $f(x) = (T(x) - g(x) h(x))/p$. Then $p$ divides the index of $\mathbb{Z}[\theta]$ in $\mathcal{O}$ if and only if $GCD(\bar{f}, \bar{g}, \bar{h})$ is not $1$.
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[1] A. Ash, J. Brakenhoff, and T. Zarrabi, Equality of Polynomial and Field Discriminants, Experiment.
Math. 16 (2007), 367–374