To explain this more precisely, consider all rational functions on our Riemann surface, in the sense of endnote 13, above. They are analogous to the rational numbers. The relevant Galois group is defined as the group of symmetries of a number field obtained by adjoining solutions of polynomial equations such as x^2 = 2 to the rational numbers. Likewise, we can adjoin solutions of polynomial equations to rational functions on a Riemann surface X. It turns out that when we do this, we obtain rational functions on another Riemann surface X′, which is a “covering” of X; that is, we have a map X′ → X with finite fibers. In this situation, the Galois group consists of those symmetries of X′ which leave all points of X unchanged. In other words, these symmetries act along the fibers of the map X′ → X.
Now observe that if we have a closed path on the Riemann surface X, starting and ending at a point P on X, we can take each point of X′ in the fiber over P and “follow” it along this path. When we come back, we will in general get a different point in the fiber over P, so we obtain a transformation of this fiber. This is the phenomenon of monodromy, which will be discussed in more detail in Chapter 15. This transformation of the fiber may be traced to an element of the Galois group. Thus, we obtain a link between the fundamental group and the Galois group.
[amor y matemáticas, pg. 511]