Conjecturally, every finite group is the Galois group of some extension of the rationals. This question made me wonder what is known about infinite simple groups occurring as Galois groups.

*What are the infinite simple groups that are expected to be Galois groups, i.e., profinite?* Are they classified? Are there any examples of such extensions?

algebraicextension. I have some notes in which I suggest a definition of Galois transcendental extensions, and then $\mathbb{C}/\overline{\mathbb{Q}}$ is Galois and its automorphism group is indeed a huge simple group. But it's not a profinite group, of course.) – Pete L. Clark Jan 29 '11 at 14:58