Fix natural numbers $d,N$ and a polynomial $\Delta \in \mathbb{C}[x\_1,\ldots,x\_d]$. Let $S\_{d,N}$ be the set of field extensions $K/ \mathbb{C}(x\_1,\ldots,x\_d)$ such that


1. The degree $[K: \mathbb{C}(x\_1,\ldots,x\_d)]$ is bounded by $N$.
2. The discriminant of $K/ \mathbb{C}(x\_1,\ldots,x\_d)$ is $\Delta$.
3. $K$ is generated by an element whose minimal polynomial has coefficients in $\mathbb{C}[x\_1,\ldots,x\_d]$ that have degrees at most $N$.


Question: Is $S\_{d,N}$ finite?



Finiteness of number of extensions with bounded degree and discriminant
Yes, and I don't think you need 3. Let $D\subset \mathbb{C}^d$ be the locus where $\Delta $ vanishes. You are looking at étale covers of a fixed degree $n\ (\leq N)$ of $\mathbb{C}^d\smallsetminus D$, up to birational isomorphism. Such covers are classified by an action of $\pi \_1(\mathbb{C}^d\smallsetminus D)$ on a set with $n$ elements. Since $\pi \_1(\mathbb{C}^d\smallsetminus D)$ is finitely generated there are only finitely many possibilities.


