TITLE: Does every ring of integers sit inside a monogenic ring of integers?
QUESTION [51 upvotes]: Given a number field $K/\mathbf{Q}$ whose ring of integers $\mathcal{O}_K$ is, in general, not of the form $\mathbf{Z}[\alpha]$ (not monogenic), does there exist an extension $L/K$ which has $\mathcal{O}_L= \mathbf{Z}[\alpha]$ (is monogenic)?
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As noted by Jake: Kronecker-Weber implies this is true when $K/\mathbf{Q}$ is abelian.

REPLY [2 votes]: [edit: initially I thought there is no local obstruction but now I'm not sure about that so I deleted what I wrote on that]
Although I don't have proof, I suspect that most fields $K$ are counter examples to the OP's question. Empirically, most number fields are not monogenic. If you compute the best generator that polredabs in GP/PARI can find, and find that it does not generate $O_K$, as is almost always the case (except for low degree or cyclotomic, etc), then it is very likely that $O_K$ is not monogenic. If $O_K$ is monogenic but polredabs didn't find a generator for $O_K$, then $O_K$ would have a generator that is much better than the one found by polredabs.  Now polredabs in GP/PARI is based on LLL techniques, and while one can not prove that LLL finds the best element, it is only a bounded factor worse than optimal. This means that the empirical observation that polredabs usually doesn't find a generator for $O_K$ is good empirical evidence that most of these $O_K$ are not monogenetic. It is hard to see how increasing the degree could help globally, so it seems likely that for most fields $K$ (other than very special cases) there will not be an extension $L$ for which $O_L$ is monogenic.