Currently, I am trying to understand the following theorem with its proof (taken from Milne's Fields and Galois Theory, page 70):
As you can see, I am questioning the well-definedness of the sum for the case when $G$ is infinite.
Question: Did the author implicity assumed that $G$ is finite or is this argument still valid for an infinite Galois group $G$?
Also, or those of you who are curious what Corollary 5.15 is:
