Let
N
(
n
,
A
n
,
X
)
N(n, A_n, X)
be the number of number fields of degree
n
n
whose Galois closure has Galois group
A
n
A_n
and whose discriminant is bounded by
X
X
. By a conjecture of Malle, we expect that
N
(
n
,
A
n
,
X
)
∼
C
n
⋅
X
1
2
⋅
(
log
X
)
b
n
N(n, A_n, X)\sim C_n\cdot X^{\frac {1}{2}} \cdot (\log X)^{b_n}
for constants
b
n
b_n
and
C
n
C_n
. For
6
≤
n
≤
84393
6 \leq n \leq 84393
, the best known upper bound is
N
(
n
,
A
n
,
X
)
≪
X
n
+
2
4
N(n, A_n, X) \ll X^{\frac {n + 2}{4}}
, by Schmidt’s theorem, which implies there are
≪
X
n
+
2
4
\ll X^{\frac {n + 2}{4}}
number fields of degree
n
n
. (For
n
>
84393
n > 84393
, there are better bounds due to Ellenberg and Venkatesh.) We show, using the important work of Pila on counting integral points on curves, that
N
(
n
,
A
n
,
X
)
≪
X
n
2
−
2
4
(
n
−
1
)
+
ϵ
N(n, A_n, X) \ll X^{\frac {n^2 - 2}{4(n - 1)}+\epsilon }
, thereby improving the best previous exponent by approximately
1
4
\frac {1}{4}
for
6
≤
n
≤
84393
6 \leq n \leq 84393
.