  How often is a quintic polynomial solvable by radicals? We establish that the
number of such polynomials, monic and irreducible with integer coefficients in
$[-H,H]$, is $O(H^{3.91})$. More generally, we show that if $n \ge 3$ and $n
\notin \{ 7, 8, 10 \}$ then there are $O(H^{n-1.017})$ monic, irreducible
polynomials of degree $n$ with integer coefficients in $[-H,H]$ and Galois
group not containing $A_n$. Save for the alternating group and degrees
$7,8,10$, this establishes a 1936 conjecture of van der Waerden.
