October 23rd
Today I learned a reason that monogenic extensions are, in a sense, kind of rare. Fix $K$ a number field. In a sense, for some $\alpha$ with minimal polynomial $f(x),$ the index $[\mathcal O_K:\ZZ[\alpha]]$ provides a measuring tool for how far away from a monogenic extension we are, and the same index describes the discrepancy between the Dedekind-Kummer factorization algorithm for ideals and actual prime-splitting. Namely, the condition that $p\nmid[\mathcal O_K:\ZZ[\alpha]]=1$ would be always satisfied in a monogenic extension, so we're always safe. But the Dedekind-Kummer factorization algorithm is often too nice to properly work.
For example, if $(2)$ splits completely in $\mathcal O_K$ where $K$ is not a quadratic field, then we immediately get a contradiction. Indeed, by Dedekind-Kummer says that $(2)$ splitting completely is the same as $f(x)\pmod2$ splitting completely, but there are only two possible roots for $f(x)$ to have, so splitting completely just isn't possible.
I guess I should provide a more concrete example. As a lemma to quadratic reciprocity, we show that for prime $n\mid p-1,$ $q$ is an $n$th power$\pmod p$ if and only if $q$ splits splits completely in the field of degree $n$ between $\QQ$ and $\QQ(\zeta_q).$ So, $(2)$ will split completely in the cubic subfield of $\QQ(\zeta_p)$ if and only if $2$ is a cube$\pmod p,$ where $p\equiv1\pmod3.$ Say, $4^3\equiv2\pmod{31}$ implies that the cubic subfield of $\QQ(\zeta_{31})$ named $F_3$ provides an example of a non-monogenic extension.
I'm not currently sure how to extract this subfield. Alec suggested considering the trace of some element because we know the Galois group. Indeed, note that\[\sum_{k=0}^{30}\zeta_{31}^{k^3}\]will be preserved by the Galois subgroup of $F_3,$ which is corresponds to the set of cubes in $(\ZZ/31\ZZ)^\times.$ Sage says that the minimal polynomial of this thing is $x^3 - 93x - 124,$ which is not at all clear to me, but sure. It follows that\[F_3=\QQ(\alpha),\qquad\alpha^3-93\alpha-124=0.\]That's a concrete field, but it still doesn't feel super satisfying because I didn't really generate it—I just handed it to Sage, and Sage will do the work.