We propose a framework to prove Malle's conjecture for the compositum of two number fields based on proven results of Malle's conjecture and good uniformity estimates. Using this method, we prove Malle's conjecture for $S_n\times A$ over any number field $k$ for $n=3$ with $A$ an abelian group of order relatively prime to 2, for $n= 4$ with $A$ an abelian group of order relatively prime to 6, and for $n=5$ with $A$ an abelian group of order relatively prime to 30. As a consequence, we prove that Malle's conjecture is true for $C_3\wr C_2$ in its $S_9$ representation, whereas its $S_6$ representation is the first counter-example of Malle's conjecture given by Klüners. We also prove new local uniformity results for ramified $S_5$ quintic extensions over arbitrary number fields by adapting Bhargava's geometric sieve and averaging over fundamental domains of the parametrization space.

我们提出了一个框架，以证明的Malle猜想的结果和良好的均匀性估计为基础，对两个数域的组合证明Malle的猜想。使用这种方法，我们证明马勒对猜想$ S_N \次甲$在任意数量的字段$ķ$为$ n = 3的$与$ A $的顺序相对素2的阿贝尔群，对于$ n = 4的$与$ A $的一个素数阶的素数组相对于6，并且对于$ n = 5 $和$ A $的一个素数阶的素数组相对于30。因此，我们证明了Malle的猜想对于$ C_3 \ wr C_2是正确的$以其$ S_9 $表示形式，而其$ S_6 $表示形式是Klüners给出的Malle猜想的第一个反例。我们还通过适应Bhargava的几何筛网并对参数化空间的基本域求平均，证明了任意数域上$ S_5 $五次扩展的五次扩张的新的局部均匀性结果。