# nLab number field

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

A number field is a finite field extension of the field of rational numbers, $ℚ$, In other words, a field $k$ of characteristic zero such that under the field homomorphism $i:ℚ↪k$, the field $k$ is a finite-dimensional vector space over $ℚ$ with respect to the scalar multiplication action of $ℚ$

$ℚ\otimes k\stackrel{i\otimes 1}{\to }k\otimes k\stackrel{\mathrm{mult}}{\to }k$\mathbb{Q} \otimes k \stackrel{i \otimes 1}{\to} k \otimes k \stackrel{mult}{\to} k

on the underlying additive group of $k$.

## Applications

Number fields are the basic objects of study in algebraic number theory. For example, one is typically interested in the arithmetic structure of $k$, including for example the structure of the ring of algebraic integers ${𝒪}_{k}$ in $k$, the decomposition of primes in $ℤ$ in terms of prime ideals in ${𝒪}_{k}$, the structure of the unit group of ${𝒪}_{k}$, the structure of the ideal class group?, the detailed study of the zeta function of $k$, and much more.

## Properties

Number fields $k$ are examples of global field?s, in fact they are the global fields of characteristic zero. They are often studied in terms of how they embed in their rings of adeles? ${𝔸}_{k}$, which are built from the local completions of $k$.

Revised on November 7, 2013 01:58:08 by Urs Schreiber (188.200.54.65)