Let $K_q$ denote the unique quadratic subextension of the ray class field over $\mathbb{Q}$ of conductor $q\times\infty$. Then $K_q$ should be $\mathbb{Q}(\sqrt{q})$ if $q$ if 1 mod 4 and $\mathbb{Q}(\sqrt{-q})$ if $q$ is 3 mod 4 if I'm not mistaken. I've shown if $p$ splits in $K_q$ then $p$ is a square mod $q$ (without QR) and I need to show the converse without using quadratic reciprocity. (I need to generalize it to a setting where I haven't proven the exact quadratic reciprocity law I believe exists). Here $q$ and $p$ are both primes and we can assume $q$ is odd.

The question in the title would show this for $q=3$.