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Let $K$ be a number field and $k\geq 2$ be an integer. Let $(n_1,n_2, \dots,n_k)$ be a vector with entries $n_i\in \mathbb{Z}_{\geq 2}$. Given a numberfield extension $L/K$, we denote by $\widetilde{L}$ the Galois closure of $L$over $K$. We prove asymptotic lower bounds for the number of number fieldextensions $L/K$ with $[L:K]=\prod_{i=1}^k n_i$, such that$Gal(\widetilde{L}/K)$ is isomorphic to the iterated wreath product ofsymmetric groups $S_{n_1}\wr S_{n_2}\wr \dots \wr S_{n_k}$. Here, the numberfields $L$ are ordered according to discriminant$|\Delta_L|:=|Norm_{K/\mathbb{Q}} (\Delta_{L/K})|$. The results in this paperare motivated by Malle's conjecture. When $n_1=n_2=\dots =n_k$, these wreathproducts arise naturally in the study of arboreal Galois representationsassociated to rational functions over $K$. We prove our results by developingGalois theoretic techniques that have their origins in the study of dynamicalsystems.

Counting number fields whose Galois group is a wreath product of symmetric groups