According to the Wikipedia article about monodromy, the monodromy group can be defined in terms of Galois theory in following way:

Let $F(x)$ denote the field of the rational functions in the variable $x$ over the field $F$, which is the field of fractions of the polynomial ring $F[x]$. An element $y = f(x)$ of $F(x)$ determines a finite field extension $[F(x) : F(y)]$.

This extension is generally not Galois but has Galois closure $L(f)$. The associated Galois group of the extension $[L(f) : F(y)]$ is called the monodromy group of $f$.

In the case of $F = \mathbb{C} $, Riemann surface theory enters and allows for the geometric interpretation given above. In the case that the extension $[\mathbb{C}(x) : \mathbb{C}(y)]$ is already Galois, the associated monodromy group is sometimes called a group of deck transformations.

This has connections with the Galois theory of covering spaces leading to the Riemann existence theorem.

Furthermore, it is explicitly remarked that for the field $F = \mathbb{C} $, this definition of monodromy coincides with the classical one in light of complex analysis and cover theory.

My question is: how, explicitly, do these two definitions of monodromy correspond to each other for $F = \mathbb{C} $? How do we obtain a cover map giving rise to the topological monodromy action from the field extensions above?