A question about principality of ideals dividing $(p)$ in imaginary quadratic field
Solution 1:
Here's an example that makes clear what's going on. Let $d = -3 \cdot 5$; then $(-3/17) = (5/17) = -1$. This implies that $p$ splits in $k$; reducing the equation $4 \cdot 17 = x^2 + 15y^2$ modulo $5$ implies $(17/5) = +1$: contradiction by the quadratic reciprocity law.