If $K$ is a number field, is it always possible to find a finite extension $L/K$ such that $L$ is the composite of fields $L_1,\ldots, L_n$, with each $L_i$ ramified at the property that at most one finite place? prime ramifies in $L_i/\mathbb{Q}$? Equivalently, is the composite of all extensions ramified at $\leq 1$ place all of $\overline{\mathbb{Q}}$?
The first question has an affirmative answer when $K$ is abelian, but for the general case, the equivalent second question sounds too strong to be true. Any ideas?
