Given a finite Galois extension of number fields $L/K$ with Galois group $G$ and a surjection $E\twoheadrightarrow G$ of finite groups, the Galois embedding problem is the question of whether there exists a Galois extension $M/K$ containing $L$ such that $Gal(M/K)\cong E$.

Let's assume we can solve our Galois embedding problem. Are there common applications to such a result?

Do you know of an example where the solution of such a problem implied an interesting/seemingly unrelated result?

Thank you.