Let $w(x,y) \in \mathbb{C}[x,y]$ be an irreducible polynomial. The vanishing locus $Z=\{(x,y) \in \mathbb{C}^2 | w(x,y)=0 \}$ with projection to $x$-coordinate is a ramified covering $\pi : Z \to \mathbb{C}$. Lets remove ramification locus $D$ and get unramified (topological) covering $\pi' : Z' \to \mathbb{C}\backslash D$. Denote $G_1$ the group of deck transformations of this covering.
On the other hand, we have finite extension of fields $\mathbb{C}(x) \to \mathbb{C}(x)[y]/(w(x,y))$, let $G_2$ be the group of automorhisms of the extension.
For coverings and for field extensions we have notion of normality. Is this covering normal iff extension is normal?
What are relations between groups $G_1$ and $G_2$? Are they isomorphic?