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
]
[-H,H]
, is
O
(
H
3.91
)
O(H^{3.91})
. More generally, we show that if
n
⩾
3
n \geqslant 3
and
n
∉
{
7
,
8
,
10
}
n \notin \{ 7, 8, 10 \}
then there are
O
(
H
n
−
1.017
)
O(H^{n-1.017})
monic, irreducible polynomials of degree
n
n
with integer coefficients in
[
−
H
,
H
]
[-H,H]
and Galois group not containing
A
n
A_n
. Save for the alternating group and degrees
7
,
8
,
10
7,8,10
, this establishes a 1936 conjecture of van der Waerden.