Suppose we have a ramified covering of Riemann surfaces, of finite degree . The function field of is the field of meromorphic functions . Now, given , we have the contravariant functor , which takes to . This functor allows us to consider as a subobject of , i.e., as a field extension of . Now I've heard that, as long as the degree of the covering is finite, then is an algebraic extension of , and . This idea is very beautiful, and I've been told it's the basic idea of Grothendieck's Galois theory. It seems "obvious" to me that this is true, but I'm clueless at how one would go about proving it.
Any insight is much appreciated!