---
tag: H5RC67SD24
kind: concept
name: Ramification
msc: [11S15, 11R04]
sources:
  - url: https://en.wikipedia.org/wiki/Ramification_(mathematics)
---

## 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|E0VZTNPJXA]] of $L$.
Ramification is one part of the [[splitting of prime ideals|B7KNZCBFYT]], and the word comes from the branching of covering maps of Riemann surfaces.

## Explanations

### L1 human

When a prime number is factored in a larger ring of integers, the factors may come with exponents bigger than $1$.
This is ramification: two or more factors have merged into one.
In $\Z[i]$, the prime $2$ factors as $-i(1+i)^2$, so $2$ ramifies with index $2$.
Every other prime number is unramified in $\Z[i]$.
A picture: the map $z \mapsto z^2$ on the complex plane is two-to-one except at $0$, where the two preimages come together; this point is a branch point.