Embedding problems with cyclic kernel of order 4
This paper considers Galois embedding problems with kernel \(C_4\) cyclic of order 4, over a field of characteristic not 2. In particular, if \(M/K\) is a finite Galois extension with Galois group \(G\) and \[ 1 \to C_4 \to E \to G \to 1 \] is a group extension, one asks whether there exists a Galois extension \(F/K\) with \(M \subseteq F\) and an isomorphism (or monomorphism) \(\phi\) from the Galois group of \(F/K\) to \(E\) such that the composition of the projection \(E \to G\) with \(\phi\) is the natural restriction from the Galois group of \(F/K\) to the Galois group of \(M/K\). It is shown that such an embedding problem with kernel \(C_4\) can be constructively reduced to two embedding problems where the kernels are groups of roots of unity. These embedding problems are called Brauer type embedding problems and have the advantage that they have a nice criterion for solvability in terms of crossed product algebras. These results in turn are used to obtain criteria for embedding cyclic extensions of degree 4 in cyclic extensions of degree 16, and for embedding dihedral extensions of degree 8 in dihedral, quasi-dihedral, and quaternionic extensions of degree 32. Finally, these results are applied to obtain some automatic realizability results \(G \Rightarrow H\), where any field admitting a \(G\)-extension must also admit an \(H\)-extension. It is shown that \(Q_{32} \Rightarrow D_{16}\) and \(D_8 \Rightarrow D_{16}\) or \(Q_{16}\), where \(D_{2n}\) denotes the dihedral group of order \(2^{n+1}\) and \(Q_{2n}\) denotes the quaternionic group of order \(2^n\).
scientific article; zbMATH DE number 1216488