Definition 11.1 (Ramification). Let $0\ne 𝔭$ be a prime ideal of $O_K$, and

$$𝔭O_L=P_1^{e_1}\cdots P_r^{e_r}$$

with $P_i$ distinct prime ideals in $O_L$, and $e_i>0$.

• (i) $e_i$ is the ramification index of $P_i$ over $𝔭$.
• (ii) We say $𝔭$ ramifies in $L$ if some $e_i>1$.