Splitting of prime ideals

In a finite extension of number fields $L/K$, a nonzero prime ideal $\mathfrak{p}$ of ${\mathcal{O}}_K$ factors uniquely as $\mathfrak{p}{\mathcal{O}}_L={\prod }_i{\mathfrak{P}}_i^{e_i}$. The distinct prime ideals ${\mathfrak{P}}_i$ are the primes above $\mathfrak{p}$, the exponents $e_i$ are ramification indices, and $f_i=[{\mathcal{O}}_L/{\mathfrak{P}}_i:{\mathcal{O}}_K/\mathfrak{p}]$ are residue degrees. They satisfy ${\sum }_i{e}_i{f}_i=[L:K]$.

A prime splits completely if it has $[L:K]$ distinct factors, all with $e_i=f_i=1$. In the Gaussian integers, $(5)=(2+i)(2-i)$ splits, $(3)$ is an inert prime, and $(2)$ ramifies because $2=-i(1+i{)}^2$. The three behaviors distinguish factorization of ideals from factorization of ordinary integers.