---
date: 2022-04-05 23:42
modification date: Tuesday 5th April 2022 23:42:25
title: "number field"
aliases: [number field]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #NT/algebraic 
- Refs:
	- #todo/add-references
- Links:
	- [CM field](Unsorted/CM%20field.md)

---

# number field

Any finite extension $K/\QQ$, so $[K: \QQ] < \infty$.

Can take [ring of integers](Unsorted/ring%20of%20integers.md) $\OO_K$ as the [integral closure](Unsorted/integrally%20closed.md) of $\ZZ$ in $K$, or equivalently the [algebraic](algebraic) integers in $K$. Note $\ff(\OO_K) = K$ and $\OO_K$ is a [Dedekind domain](Dedekind%20domain.md) of [Krull dimension](Unsorted/Krull%20dimension.md) one.

Can always find a power basis by the [primitive element theorem](primitive%20element%20theorem).

Can define [trace](Unsorted/trace%20(monoidal%20categories).md) and [norm](Unsorted/field%20norm.md) of $x$ as the trace and determinant of the map $m_x: a\mapsto ax$.

Has [archimedean](Unsorted/absolute%20value.md) [places](Unsorted/Valuations.md) and nonarchimedean places.

See also [ramification](Unsorted/ramification%20index.md), [inertia group](Unsorted/ramification%20index.md)