---
tag: MWBDFS2KJ5
kind: concept
name: Algebraic number field
msc: [11R04, 11R21]
specializes: [04C5ZBG1WT, 634XKY8RZ0]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_number_field
same_as:
  wikidata: Q616608
  wikipedia: Algebraic number field
---

## Summary

An algebraic number field, or number field, is a [[field extension|0EQWVHF3W5]] of the rational numbers of finite degree.
Number fields are the main objects of algebraic number theory.
Each number field $K$ has a ring of integers $\mathcal{O}_K$, a [[Dedekind domain|7NPY2CN5KB]] that plays the role of $\Z$ inside $\Q$.
The failure of unique factorisation in $\mathcal{O}_K$ is measured by the [[ideal class group|WQ46BD3ARY]].

## Explanations

### L1 human

Start with the fractions $\Q$ and adjoin a root of a polynomial, for example $i = \sqrt{-1}$ or $\sqrt{2}$.
The numbers you can then form with $+$, $-$, $\times$ and $\div$ make up a number field.
For example, $\Q(i)$ consists of the numbers $a + bi$ with $a, b$ rational.
It is a two-dimensional space over $\Q$, with basis $1, i$.
Number theory in such fields explains facts about ordinary integers, such as which primes are sums of two squares.