  Given a prime $p$, a number field $\K$ and a finite set of places $S$ of
$\K$, let $\K_S$ be the maximal pro-$p$ extension of $\K$ unramified outside
$S$. Using the Golod-Shafarevich criterion one can often show that $\K_S/\K$ is
infinite. In both the tame and wild cases we construct infinite subextensions
with bounded ramification using the refined Golod-Shafarevich criterion. In
  the tame setting we achieve new records on Martinet constants (root
discriminant bounds) in the totally real and totally complex cases.
  We are also able to answer a question of Ihara by producing infinite
asymptotically good extensions in which infinitely many primes split
completely.
