# Relationships between the field of meromorphic functions and the fundamental group

Let $X$ be a connected compact Riemannian surface. Then the ring of meromorphic functions on $X$ is a field and we denote it as $M(X)$. Let $E$ be a finite extension over $M(X)$. Then we can show that there is a corresponding $Y\to X$ restricts to a covering of $X\setminus\{P_1,\ldots,P_n\}$. My question is if $E$ is a Galois extension then whether I can say that the topology fundamental group of $Y'$ is a normal subgroup of that of $X'$ and whether I can say $\operatorname{Gal}(E/M(X))\cong\pi_1(X')/\pi_1(Y')$ (we assume $X',Y'$ are both connected so the base point is omitted).

• $X' = X \setminus \{P_1 \ldots P_n \}$ ? – mercio Jan 1 '17 at 9:37
• Yes. It is a complement of a finite sets. – Li Li Jan 1 '17 at 9:41