Upper bound on the number of extensions of a given number field

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.