Construction of \(2^mS_n\)-fields containing a \(C_{2^m}\)-field (Q1268096)

The author considers the embedding problem given by the epimorphism \(2^m{\mathfrak S}_n^+\to{\mathfrak S}_n\). Here \({\mathfrak S}_n\) denotes the symmetric group on \(n\) letters and \({\mathfrak S}_n^+\) is one of the two double covers of \({\mathfrak S}_n\). She obtains a criterion for the solvability of the embedding problem and a method of computation of the solutions by reducing to an embedding problem with kernel of order 2. Under certain conditions on the ground field the obstruction is given in terms of quaternion algebras and Hasse-invariants of the trace form of a subextension. The proofs are based on the author's previous work and results of Fröhlich, Quer, Serre and Swallow.