  We determine the density of integral binary forms of given degree that have
squarefree discriminant, proving for the first time that the lower density is
positive. Furthermore, we determine the density of integral binary forms that
cut out maximal orders in number fields. The latter proves, in particular, an
``arithmetic Bertini theorem'' conjectured by Poonen for
$\mathbb{P}^1_\mathbb{Z}$.
  Our methods also allow us to prove that there are $\gg X^{1/2+1/(n-1)}$
number fields of degree~$n$ having associated Galois group~$S_n$ and absolute
discriminant less than $X$, improving the best previously known lower bound of
$\gg X^{1/2+1/n}$.
  Finally, our methods correct an error in and thus resurrect earlier
(retracted) results of Nakagawa on lower bounds for the number of totally
unramified $A_n$-extensions of quadratic number fields of bounded discriminant.
