Ramification

Remarks

For number fields the residue field extensions are finite extensions of finite fields, hence separable, so no separability condition is needed. A prime number $p$ ramifies in a number field $L$ if and only if $p$ divides the discriminant $d_L$; in particular only finitely many primes ramify. See splitting of prime ideals for the full description of the factorisation.

Example: in $L = \Q(\sqrt{-5})$, the ideal $2\mathcal{O}_L$ equals $\mathfrak{P}^2$ with $\mathfrak{P} = (2, 1 + \sqrt{-5})$. So $e(\mathfrak{P} \mid 2) = 2$ and $2$ is ramified, in line with $2 \mid d_L = -20$.

Non-example: $3$ is unramified in $\Q(\sqrt{-5})$. Indeed $x^2 + 5 \equiv (x - 1)(x + 1) \pmod 3$ has distinct roots, so $3\mathcal{O}_L = (3, \sqrt{-5} - 1)(3, \sqrt{-5} + 1)$ with two distinct primes, each of exponent $1$; and $3 \nmid 20$.