Ah, I believe it makes sense now...
1. The extension is finite for otherwise there would be an infinite number of automorphisms of K(X) that fix K (Correct me if I'm wrong, but one way to see this would be to pick a basis for K(X) over K, and then determine an automorphism by specifying a...
The claim is:
Let K be a subfield of the complex numbers, and let X be a finite set of complex numbers. If the elements of X are permuted by any automorphism of the complex numbers that fixes K, then the field obtained by adjoining the elements of X to K is a finite, Galois extension of K...