  We study the number $N(n, A_n, X)$ of number fields of degree $n$ whose
Galois closure has Galois group $A_n$ and whose discriminant is bounded by $X$.
By a conjecture of Malle, we expect that $N(n, A_n, X) \sim C_n X^{1/2} (\log
X)^{b_n}$, for constants $b_n$ and $C_n$. For $5 < n < 84394$, the best known
upper bound is $N(n, A_n, X) \ll X^{\frac{n + 2}{4}}$; this bound follows from
Schmidt's Theorem, which implies there are $\ll X^{\frac{n + 2}{4}}$ number
fields of degree $n$. (For $n > 84393$, there are better bounds due to
Ellenberg and Venkatesh.) We show, using the important work of Pila on counting
integral points on curves, that $N(n, A_n, X) \ll X^{\frac{n^2 - 2}{4(n -
1)}+\epsilon}$, thereby improving the best previous exponent by approximately
1/4 for $5 < n < 84394$.
