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?

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?