For instance my reference was Rudin, Real and complex analysis, in which he defines region as above, and for him a domain is the domain of a function(as opposed to co-domain). I suggest that this question be closed because it is too vague.

The definition I have read of region is that it is a connected open set in the complex plane. This is matter of your choice of complex analysis book. I do not think you should spend much time on this matter.

I suppose it is in this Smith normal form that you prove the crucial part that $a_i | a_{i +1}$.

@Emerton. Your answer was extremely helpful to me, and it was very enlightening to read it. However I am accepting YBL's answer since I had asked for references and he gave me some. I hope you don't mind.

I suppose for transcendental extensions Artin's theorem on linear independence of characters will break down, and without this powerful theorem it will be hard to draw any consequences. Are there deeper reasons for you getting interested in transcendental Galois extensions, in addition to it being a curiosity?

@Matus. To strengthen my argument in my answer, I have found that ergodic theory will run into problems without a good notion of "almost nowhere". Please see my answer again.

Thanks for the explanation. So if I understood correctly, Artin motives introduce a geometric Galois theory, suitably integrated into a bigger picture according to the visions of Grothendieck.

The last chapter of the new edition of Courant and Robbins' "What is Mathematics?" has an appendix in which Ian Stewart gives a sketch of nonstandard analysis and spectacular applications to proofs in analysis.

@Pete. You did not slip up. There was not enough information available for you. In any case, P. R. Halmos gives an argument in his autobiography that "he" is also the neuter pronoun and its use is perfectly acceptable in instances where the gender is unknown. And he considers it better than "he or she", "(s)he", "s/he" etc..

I had also mentioned him; but had to take it down by the objections of Jonas Meyer!

Ok, very well. Your objection is valid. However I am reluctant to further shorten my answer to just one sentence. So let it stand!

This is a good formulation of what I wanted to say.

@Jonas Meyer. Thanks. Shortened the answer accordingly.