Let $X$ be a connected, integral curve over a field $k$, and let $Y \rightarrow X$ be a finite etale cover. Corresponding to this cover there is a finite extension of function fields $k(Y): k(X)$. 


Question: is it possible to characterize the finite extensions of $k(X)$ which arise in this way in a purely field theoretic manner? In other words, given any finite $F: k(X)$, is there some field theoretic property that $F$ satisfies iff $F=k(Y)$ for some finite etale $Y \rightarrow X$?


Feel free to assume whatever else you'd like about $X$ if it helps.



Recognizing etale covers on the level of function fields
Yes, this is possible. For simplicity, compare first to number theory: given a finite field extension $E / \mathbf Q\_p$ you want to know whether $\mathcal O\_E / \mathbf Z\_p$ ($\mathcal O\_E$ is the integral closure of $\mathbf Z\_p$ in $E$) is etale. This is precisely the case if the original extension is unramified (and etale, of course, so in this case just normal)


The same will work for function fields, except that you have more than one place to consider. I did not check the details, but I suppose it will be easier if $X$ is smooth proper to begin with. In this case let $Y$ be the smooth curve that corresponds to $F$ by the bijection of function fields vs. smooth proper curves. Then, replace $\mathcal O\_E$ above by the valuation rings corresponding to all the closed points of $Y$. Because of the smoothness assumption, these rings will be DVR's, so the question whether the extension of the local DVR's $\mathcal O\_y$ of $Y$ and $X$ is etale is determined by studying the splitting of the (ideal generated by) a uniformizing element $\pi \mathcal O\_y$ (here $\pi \in \mathcal O\_x$ is the uniformizer = generator of the maximal ideal).


