TITLE: Quadratic field such that a certain finite set of primes split
QUESTION [5 upvotes]: Given a finite set $S$ of primes, is it possible to find an imaginary quadratic field $K$ such that all primes in $S$ are split completely in $K$?

REPLY [3 votes]: Sure. Let me just assume WLOG that $S$ contains $2$. Let $D$ be squarefree. If $D \equiv 1 \bmod 4$, then $\mathbb{Q}(\sqrt{D})$ has ring of integers $\mathbb{Z}[x]/(x^2 - x - \frac{D-1}{4})$. Then $2$ splits if and only if $D \equiv 1 \bmod 8$. An odd prime $p$ splits if $D \equiv 1 \bmod p$, so it suffices to find a (negative) squarefree $D \equiv 1 \bmod 8$ such that $D \equiv 1 \bmod p$ for all odd primes $p \in S$. In fact a prime with this property exists by Dirichlet's theorem.