The most effective challenge to the inverse problem of Galois theory has been Hilbert's Irreducibility Theorem. Indeed, one may use both arithmetic and geometry in order to realize finite groups over Q(t). Once this has been successfully done for a finite group G, HIT yields many specializations of t to elements of Q that lead to a realization of G over Q. However, the realization of G over Q(t) usually requires the existence of a rational point on a certain algebraic variety defined over Q. Unfortunately, one cannot always guarantee the existence of such a point, so the inverse Galois problem over Q is still wide open.
The only known class of fields for which points that lead to realization of all finite groups exist is that of ample fields. A field K is said to be ample if every absolutely irreducible curve C defined over K with a K-rational simple point has infinitely many K-rational points. Among others, PAC fields, Henselian fields, and real closed fields are ample. Using a method called algebraic patching we will prove that if K is an ample field, then every finite split embedding problem over K(t) is properly solvable. In particular, if K is countable and algebraically closed, this implies that Gal(K) is isomorphic to the free profinite group F on countably many generators. Also, if K is PAC and countable, then K is Hilbertian if and only if Gal(K) is isomorphic to F.
Last modification: 16 February 2012.