Splitting of rational primes in a quadratic field
= Splitting of rational primes in a quadratic field
For a quadratic field of discriminant $D$ and a rational prime $p$, the ideal $(p)$ splits into two distinct prime ideals, remains prime, or ramifies according as the quadratic character $\chi_D(p)$ is $1$, $-1$, or $0$. The corresponding prime-ideal norms are $(p,p)$, $p^2$, and $p$, respectively.