Given a "random" polynomial over the integers, it is expected that, with high probability,
it is irreducible and has a big
Galois group over the rationals. Such results have been long known when the degree is bounded
and the coefficients
are chosen uniformly at random from some interval, but the case of bounded coefficients and unbounded degree
remained
open. Very recently, Emmanuel Breuillard and Peter Varju settled the case of bounded coefficients conditionnally
on the
Riemann Hypothesis for certain Dedekind zeta functions. In this talk, I will present unconditional
progress towards this problem,
joint with Lior Bary-Soroker and Gady Kozma.