### Course Description: Algebraic Patching

#### Lecturer: Moshe Jarden

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.

Lecture Notes

Last modification: 16 February 2012.