Splitting of prime ideals in Galois extensions

In a Galois extension, the prime ideals above a fixed unramified prime have a common ramification index and residue degree, and the Galois group acts transitively on them.

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.

A prime $p$ ramifies in a number field when its prime-ideal factorization contains a factor with exponent greater than one.

A finite extension of number fields is unramified everywhere when every finite prime ideal is unramified. If infinite places are included in the convention, one also requires every real place to remain real.

If $p\equiv 11(\mathrm{mod}12)$ and $K=\mathbb{Q}(\sqrt{-p})$, then ${\mathcal{O}}_K=\mathbb{Z}[(1+\sqrt{-p})/2]$. The generator's minimal polynomial reduces modulo $3$ to $T(T-1)$, so $(3)$ splits into two distinct prime ideals of norm three.