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 subset of K subset of L with L/F finite and separable, we explicitly construct an embedding of the Galois group Gal(L-c/F) into the regular wreath product Gal(L-c/K-c)((sic)r)Gal(K-c/F) . Here Lc (resp. K-c) denotes the Galois closure of L/F (resp. K/F). Similarly, we also construct an explicit embedding of the Galois group Gal(L-c/F) into the smaller sized wreath product Gal(L-c/K)((sic)Omega)Gal(K-c/F) , where Omega=Hom(F)(K,K-c) is acted on by composition of automorphisms in Gal(K-c/F). Moreover, when L/K is a Kummer extension we prove a sharper embedding, that is, that Gal(L-c/F) embeds into the wreath product Gal(L/K)((sic)Omega)Gal(K-c/F) . As corollaries we obtain embedding theorems when L/K is cyclic and when it is quadratic with char(F)not equal 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.