  We determine the density of monic integer polynomials of given degree $n>1$
that have squarefree discriminant; in particular, we prove for the first time
that the lower density of such polynomials is positive. Similarly, we prove
that the density of monic integer polynomials $f(x)$, such that $f(x)$ is
irreducible and $\mathbb Z[x]/(f(x))$ is the ring of integers in its fraction
field, is positive, and is in fact given by $\zeta(2)^{-1}$.
  It also follows from our methods that there are $\gg X^{1/2+1/n}$ monogenic
number fields of degree $n$ having associated Galois group $S_n$ and absolute
discriminant less than $X$, and we conjecture that the exponent in this lower
bound is optimal.
