Subgroups of \(\text{Hol} Q_ 8\) as Galois groups
Let \(p\) be a prime and let \(L/K\) be a Galois extension of fields of characteristic \(\neq p\). The main theorem of the paper under review states that the solvability of the embedding problem given by the non-split group extension \[ 1 \to C_p \to E \to G(L/K) \to 1 \] with kernel of order \(p\) is equivalent to the solvability of a central embedding problem over a field \(K' \supset K\) which contains the \(p\)-th roots of unity. The reduction is effective. All solutions of the first embedding problem can be obtained from a single solution of the latter embedding problem. Finally the author applies his reduction to compute the obstruction to certain embedding problems, where \(E\) is one of four subgroups of the holomorph \(\text{Hol}(Q_8) \simeq Q_8 \rtimes \Aut (Q_8) \simeq Q_8 \rtimes S_4\) of the quaternion group \(Q_8\) including the holomorph itself.
scientific article; zbMATH DE number 883770