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Suppose $k$ is an algebraically closed field with characteristic 0 and let $X \subset \mathbb{A}^n$ be an irreducible curve.

If $f \colon X \to \mathbb{A}^1$ is finite of degree $d$, then the coordinate ring $k[X]$ is an integral extension of $f^\ast( k[\mathbb{A}^1]) = k[f]$. Following this question, we say $k[X]$ is monogenic over $k[f]$ if there is an $\alpha \in k[X]$ such that $k[X]= k[f][\alpha]$.

It appears that not every number field is monogenic (over $\mathbb{Z}$) and that, in general, it is difficult to decide if a given primitive element $\alpha$ of a number field $K$ generates the ring of integers $\mathcal{O}_K$ over $\mathbb{Z}$. However, this paper provides several criteria for making such a decision.

Is the situation similar for $k[X]$ over $k[f]$? That is, are there examples of curves and finite maps such that $k[X]$ is not monogenic over $k[f]$?

If so, does the property of monogenicity have a nice geometric interpretation?

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    $\begingroup$ Yes, there is a geometric interpretation Indeed, $k[X]$ is monogenic over $k[f]$ if and only if $X$ admits a closed immersion into $\mathbb A^2$ whose $x$ coordinate is $f$. (The equivalence proceed by taking $\alpha$ to be the $y$ coordinate). $\endgroup$
    – Will Sawin
    Commented Feb 26, 2017 at 21:30
  • $\begingroup$ @WillSawin, thank you very much - that was a surprisingly simple answer. $\endgroup$
    – charlie katerba
    Commented Feb 26, 2017 at 22:30

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