论文标题
在给定数字字段的扩展数量上的上限
Upper bound on the number of extensions of a given number field
论文作者
论文摘要
在本文中，我们改善了数字$ n_ {k，n}（x）$ $ n $ number field $ k $的扩展的上限，并具有由$ x $的绝对判别。这是通过给出一个简短的$ \ Mathcal {o} _k $ - $ l $ $ k $的订单的基础。我们的结果概括了$ n _ {\ mathbb {q}，n}（x）$ by Lemke Oliver和Thorne上的最著名的上限。确切地说，我们证明$ n_ {k，n}（x）\ ll_ {k，n} x^{c（\ log n）^2} $对于显式常数$ c $独立于$ k $和$ n $。我们还改善了在某些连接的半神经谎言组中最大算术亚组数量的上限。
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.
