Solving embedding problems with bounded ramification

Let K / K 0be a finite Galois extension of global fields. We prove that every finite embedding problem with a solvable kernel H for K / K 0is solvable if it is locally solvable and satisfies two conditions onchar ( K 0 ) and the roots of unity in K . Moreover, the solution can be chosen to coincide with finitely many (given in advance) local solutions. Finally, and this is the main point of this work, the number of primes of K 0 that ramify in the solution field is bounded by the number of primes of K 0 that ramify in K plus the number of prime divisors of | H | , counted with multiplicity.