Let $k$ be a number field. Is it possible that $k$ has an infinite (non-abelian) extension that is unramified *everywhere*?


Thank you!



Unramified extensions of number fields
Two things: 


1) Yes, certainly. By class field theory and the finiteness of the class group, the maximal *abelian* unramified extension of *any* number field is of finite degree. Thus any infinite unramified extension is non-abelian -- in particular, any infinite class field tower. The Golod-Shafarevich examples and oodles of refinements since then all give examples.


2) Perhaps you are actually interested in non-*solvable* infinite extensions, i.e., unramified extensions which are not built up as an infinite series of unramified abelian extensions. In this case, the answer is also yes. In fact, this can even be done over number fields of class number 1. For examples, see Maire's "On Infinite Unramified Extensions."


Hope that helps.


