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.

A prime ideal of a number field splits completely in a finite extension of degree $n$ when it factors into $n$ distinct prime ideals, each of residue-field degree one. In a Galois extension, an unramified prime ideal splits completely exactly when its Frobenius conjugacy class is the identity class. The Chebotarev density theorem then gives these primes Dirichlet density $1/n$.

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.

Ramification specializes to primes as follows: 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.