TITLE: Insolvable number fields ramified only at one (small) prime
QUESTION [11 upvotes]: In his first Eilenberg Lecture at Columbia, Benedict Gross says that only recently have we been able to give examples of finite galoisian extensions $K$ of ${\bf Q}$ which are ramified only at $2$ (respectively $3$) and for which the group Gal$(K|{\bf Q})$ is not solvable.  He seemed to suggest that this was made possible by recent progress in the Langlands Programme.
Question.  Which such number fields have been discovered recently, and which bits of the Langlands Programme are needed to construct them ?

REPLY [9 votes]: The construction of such number fields is not limited to the Langlands program.  For instance, to what extent are there number fields with few ramified primes and Galois group S_n?  You won't be able to cook these out of automorphic forms.  In this connection there is a remarkable recent example of David Roberts: a number field whose Galois group is the symmetric group on 15875 letters, and whose discriminant is $-2^{130729}5^{63437}$!