On the embedding of Galois groups into wreath products

In this paper we make explicit an application of the wreath product construction to the Galois groups of field extensions. More precisely, given a tower of fields $F \subseteq K \subseteq L$ with $L/F$ finite and separable, we explicitly construct an embedding of the Galois group $\operatorname{Gal}(L^c/F)$ into the regular wreath product $\operatorname{Gal}(L^c/K^c) \wr_r \operatorname{Gal}(K^c/F)$. Here $L^c$ (resp. $K^c$) denotes the Galois closure of $L/F$ (resp. $K/F$). Similarly, we also construct an explicit embedding of the Galois group $\operatorname{Gal}(L^c/F)$ into the smaller sized wreath product $\operatorname{Gal}(L^c/K) \wr_{\Omega} \operatorname{Gal}(K^c/F)$, where $\Omega = \operatorname{Hom}_F(K, K^c)$ is acted on by composition of automorphisms in $\operatorname{Gal}(K^c/F)$. Moreover, when $L/K$ is a Kummer extension we prove a sharper embedding, that is, that $\operatorname{Gal}(L^c/F)$ embeds into the wreath product $\operatorname{Gal}(L/K) \wr_{\Omega} \operatorname{Gal}(K^c/F)$. As corollaries we obtain embedding theorems when $L/K$ is cyclic and when it is quadratic with $\operatorname{char}(F) \neq 2$. We also provide examples of these embeddings and as an illustration of the usefulness of these embedding theorems, we survey some recent applications of these types of results in field theory, arithmetic statistics, number theory and arithmetic geometry.