Automorphisms of fields and structures (Q1329600)

The main result is the following one: for any relational structure \(A = (M,R_ 1, \dots, R_ n)\) and any field \(K\) there exists a field extension \(L = L(A)\) of \(K\) such that \(\Aut (L) = \Aut (A)\) and \(K\) is contained in the field of fixed elements of \(L\); moreover the assignment \(A \mapsto L(A)\) is in a certain sense functorial. Since any group is the group of automorphisms of a certain structure, as a consequence one obtains an elementary proof of the following result of M. Dugas and R. Göbel: any infinite group is a Galois group over any field.