Cutting towers of number fields

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 are able to produce infinite asymptotically good extensions in which infinitely many primes split completely, and in which every prime has Frobenius of finite order, a phenomenon that had been expected by Ihara. We also achieve new records on Martinet constants (root discriminant bounds) in the totally real and totally complex cases.