The question as to whether there exist a positive proportion of monic irreducible integral polynomials of degree n having squarefree discriminant is an old one; an exact formula for the density was conjectured by Lenstra.  (The interest in monic irreducible polynomials f with squarefree discriminant comes from the fact that in such cases

Z[x]/(f(x)) gives the ring of integers in the number field Q[x]/(f(x)).)

In this talk, we describe recent work with Arul Shankar and Xiaoheng Wang that allows us to determine the probability that a random monic integer polynomial has squarefree discriminant - thus proving the conjecture of Lenstra.