Unramified Subextensions of Ray Class Field Towers

Fix a prime l. In this paper, we explore various senses in which the ramification in an infinite l-extension of number fields can be “small.” In particular, (1) we construct infinitely many pairs of primes p, q (distinct from l) such that admits an infinite l-extension unramified outside {p, q}; and (2) we explore the possibility of finding infinite unramified subextensions inside infinite ramified l-extensions of number fields.