TITLE: Dihedral extension unramified at primes dividing order of group?
QUESTION [5 upvotes]: Consider dihedral Galois extensions $L/\mathbb{Q}$ of degree $n$ (and we know they exist thanks to Shafarevich), can we show there always exists an extension $L/\mathbb{Q}$ unramified at $p$, for all $p \mid 2n$?

REPLY [3 votes]: Since the question mentions (the somewhat heavy machinery) "Shafarevich" as a reason for the existence of dihedral extensions, it might be worth adding that the same question has a positive answer (now, indeed due to Shafarevich's method) for all solvable groups (with the condition $p|2n$ replaced by "primes dividing the group order", or in fact even "primes in any prescribed finite set").