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To add to Keith's answer, there are various classes of number fields which are known to be not monogenic. For instance, the following paper Marie-Nicole Gras, Non monogénéité de l'anneau des entiers …

To expand slightly my brief comment. Regard $L$ as a subfield of $\mathbb{C}$. Let $L'=\mathbb{R}\cap L$. Then the index $|L:L'|=1$ or $2$. In the former case all quadratic subfields of $L$ are real. …

To expand on the excellent comments a bit, one needs both a bit more and a bit less than Chebotarev's density theorem. :-) Let's take a number field $K$, and a nonzero ideal $\mathfrak{N}$ of the rin …

The rank of the unit group of $K$ is $r+s-1$ where $r$ and $2s$ are the numbers of real and complex embeddings of $K$. As $r+2s=n$ this rank is less than $n=r+2s$, the degree of $K$. It follows that t …

Your "$x^n-\alpha$" approach is the "ramified" way to go: the extension you get localizes to a totally ramified extension of the local field $K_{\mathfrak{p}}$, which has degree $n$, so the global ext …