Relative embedding problems (Q2706615)

A relative embedding problem is given as follows: Given a Galois extension \(E_B/K\) with Galois group \(H\), suppose that there is a surjection of groups \(G\rightarrow H\). Suppose further, that some Galois subextension \(K_1/K\) of \(E_B\) with Galois group \(\bar H\) embeds in a Galois extension over \(K\) with Galois group \(G\). Then we ask whether \(E_B/K\) embeds in a Galois extension with Galois group \(G\).NEWLINENEWLINENEWLINEFor such embedding problems with abelian kernel, the authors give a reduction theorem (Theorem 3.1). Let \(L/K\) be a Galois extension with Galois group \(G\), let \(A\) be a normal abelian subgroup of \(G\), and let \(B\) be a normal subgroup of \(G\), contained in \(A \). Suppose \(E_B/K\) is a \(G/B\)-Galois extension such that \(G(L\cap E_B/K)\) is a \(G/A\)-Galois extension. Then the abelian relative embedding problem is to determine all Galois extensions \(E/K\) that extend \(E_B/K\) and such that \(G(E/K)\) is isomorphic to \(G\).NEWLINENEWLINENEWLINEThe authors illustrate their approach by considering dihedral groups \(D_n,D_m,D_k\) with \(k|m|n\).NEWLINENEWLINENEWLINEFinally, they apply these results to the theory of obstructions to central embedding problems and classify the infinite towers of metacyclic \(p\)-groups to which the reduction theorem applies.