On embedding problems with restricted ramifications. (Q1125392)

Let \(L/k\) be a finite Galois extension of algebraic number fields, with Galois group \(G\), and let \( 1\to A\to E\to G\to 1\) be a central extension of groups. The author investigates the existence of the Galois extension \(M/L/k\) such that the Galois group \(\mathrm{Gal}(M/k)\) is isomorphic to \(E\) and that \(M/L\) is unramified outside a finite set of primes of \(L\). The class number of the Hilbert \(p\)-class field is also discussed.