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 $p \mid 2n$?
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2 Answers
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$\DeclareMathOperator\Gal{Gal}$The answer is "yes", and this is an easy exercise in class field theory: if, for example, $q$ is a prime number that is $1\pmod n$, and $F$ is a quadratic number field in which $q$ splits, then there is a quotient of the ray class group of $F$ with modulus $q$ that has order $n$ and on which $\Gal(F/\mathbb{Q})$ acts by $-1$, so that the corresponding ray class field $L/F$ is Galois over $\mathbb{Q}$ with dihedral Galois group and is unramified (over $F$) outside of $q$. See, for example, Section 3.1 in https://arxiv.org/abs/0805.1231 for the details of this computation. So if you also arrange $F$ to be unramified at all $p|2n$, then $L/\mathbb{Q}$ is as required.
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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").
