Ramification

Summary

Let $L$ be a finite extension of a number field $K$. A prime ideal $\mathfrak{p}$ of $K$ ramifies in $L$ when its factorisation in $L$ has a repeated prime factor. The exponent of a prime factor is its ramification index. Only finitely many primes ramify, namely those dividing the relative discriminant. Over $\Q$ these are the primes dividing the discriminant of $L$. Ramification is one part of the splitting of prime ideals, and the word comes from the branching of covering maps of Riemann surfaces.