On embedding problems with restricted ramifications

Abstract.

Let L / k be a Galois extension with Galois group G, and $(\varepsilon ): 1\to A\to E\to G\to 1$ a central extension. We study the existence of the Galois extension M / L / k such that the Galois group Gal(M / k) is isomorphic to E and that M / L is unramified outside S, where S is a finite set of primes of L. As an application, we also study the class number of the Hilbert p-class field.