---
tag: M54ADM0CVF
kind: concept
name: Ideal norm
msc: [11R04, 13F05]
sources:
  - url: https://en.wikipedia.org/wiki/Ideal_norm
same_as:
  wikidata: Q5988007
  wikipedia: Ideal norm
---

## Summary

Let $\mathfrak{a}$ be a nonzero ideal in the ring of integers $\mathcal{O}_K$ of a [[number field|MWBDFS2KJ5]].
The norm of $\mathfrak{a}$ is the number of elements of the quotient ring $\mathcal{O}_K / \mathfrak{a}$.
It is multiplicative, and for a principal ideal it is the absolute value of the field norm of a generator.
It measures the size of an ideal.
It is used in the [[finiteness of the class number|F99RM1T02C]] and in the definition of the [[Dedekind zeta function|KXS5DQDFYD]].

## Explanations

### L1 human

In the integers, the ideal $n\Z$ has $\abs{n}$ residue classes, so its size is $\abs{n}$.
The ideal norm extends this to other number rings: it counts the residue classes modulo the ideal.
For example, in the Gaussian integers $\Z[i]$ the ideal generated by $1 + i$ has two classes, so its norm is $2$, and indeed $(1 + i)(1 - i) = 2$.
A smaller ideal has more residue classes: $4\Z$ lies inside $2\Z$ and has norm $4$ instead of $2$.