  We consider random polynomials with independent identically distributed
coefficients with a fixed law. Assuming the Riemann hypothesis for Dedekind
zeta functions, we prove that such polynomials are irreducible and their Galois
groups contain the alternating group with high probability as the degree goes
to infinity. This settles a conjecture of Odlyzko and Poonen conditionally on
RH for Dedekind zeta functions.
