Hilbert's irreducibility theorem asserts:
if f is a polynomial in two variables X,Y with integral coefficients 
that is irreducible and of degree at least 1 in Y, 
then there exists an irreducible specialization, 
i.e. a rational number a such that f(a,Y) is irreducible. 
A field with irreducible specializations is called Hilbertian.
The numerous applications of this theorem make the question of under 
what conditions an extension of a Hilbertian field is again Hilbertian 
intersting. 
It turns out the the most difficult part is separable algebraic extensions.

Jarden conjectured that if K is Hilbertian, A an abelian variety over K, 
and E/K is an extension of K that is contained in the field 
generated by all torsion points of A, then E is Hilbertian. 

In this talk I shall discuss a solution of the conjecture 
using Galois representations.