Cutting towers of number fields

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