My question refers to the proof of lemma 4.11 (page 9) from A. Kundu's "THE ETALE FUNDAMENTAL GROUP OF AN ELLIPTIC CURVE" ( math.uchicago.edu/~may/REU2017/REUPapers/Kundu.pdf)
The statement is:
Let $\phi:X \to C$ be a finite etale map of schemes. If $C$ is a non-singular curve, and $X$ is connected then $X$ is a non-singular curve.
Let $Q \in X$ and $\phi(Q)=P$, we consider the induced map of local rings $\phi_Q: O_{P,C} \to O_{Q,X}$.
By assumption $O_{P,C}$ is regular and therefore a DVR since $C$ curve. Futhermore since $\phi$ is finite and etale (= unramified + flat; see page 7).
So the goal is to show that $O_{Q,X}$ regular. Therefore it suffice to show that $O_{Q,X}$ is DVR.
One criterion is to show that it is local Dedekind domain. I'm intend to work with (DD4)) from: https://en.wikipedia.org/wiki/Dedekind_domain#Alternative_definitions:
(DD4) R is an integrally closed, Noetherian domain with Krull dimension one
Since $\phi$ is finite and $O_{P,C}$ has dimension one and Noetherian this holds also for $O_{Q,X}$.
But I don't see how do conclude that $O_{Q,X}$ is integrally closed.
Up to now we haven't used that $\phi$ is unramified; therefore $(O_{Q,X}/m_Q)= k(Q) / k(P)=(O_{P,C}/m_P)$ is finite separable extension. How does it help here?