Unique realizability of finite Abelian 2-groups as Galois groups (Q1185814)

A finite group \(G\) is called uniquely realizable if there exists a field \(K\) of characteristic 0 and exactly one Galois extension \(N/K\) with \(G(N/K)=G\). If \(K\) can be chosen to be an algebraic extension of \(\mathbb{Q}\), \(G\) is called uniquely algebraically realizable. In this paper the unique realizability property in the case of finite Abelian 2-groups is studied. Criteria for the unique realizability of a finite Abelian 2-group over a field \(K\) are given in terms of the structure of the group, the non solvability of some embedding problems and also the structure of the profinite Galois group of the maximal Abelian 2-extension of \(K\). The authors obtain a complete characterization of the finite Abelian 2-groups which are uniquely algebraically realizable. In the general case, they give a sufficient condition for the unique realizability.