  In this paper we improve the upper bound of the number $N_{K, n}(X)$ of
degree $n$ extensions of a number field $K$ with absolute discriminant bounded
by $X$. This is achieved by giving a short $\mathcal{O}_K$-basis of an order of
an extension $L$ of $K$. Our result generalizes the best known upper bound on
$N_{\mathbb{Q}, n}(X)$ by Lemke Oliver and Thorne to all number fields $K$.
Precisely, we prove that $N_{K, n}(X) \ll_{K, n} X^{c (\log n)^2}$ for an
explicit constant $c$ independent on $K$ and $n$. We also improve the upper
bound of the number of maximal arithmetic subgroups in certain connected
semisimple Lie groups.
