nLab
number field

Definition

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

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

Related concepts

• algebraic curve