Transcendental Galois Theory

DISCLAIMER: This is just a draft. I am not sure of the final purpose of this document, so at the moment I have included lots of known results, in particular proofs of the usual Fundamental Theorem of Galois theory (in the finite and then the algebraic case). Seeing such proofs helps (me) to understand the analogies between the Fundamental Theorem presented here and the usual cases. Also, CAVEAT EMPTOR. Many of the assertions here do not appear with complete proofs. The proofs are pretty routine, except for one: at the moment I am just hoping that the claim about Galois closures of genus zero coverings is true. This claim is used to prove what is the most interesting result in the paper: namely, that a transcendental extension K/F is Galois only if K is algebraically closed. Please let me know if you have any ideas about how to prove this claim, and/or how to prove the deduction without it!