nLab
number field

Contents

Definition

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

ki1kkmultk\mathbb{Q} \otimes k \stackrel{i \otimes 1}{\to} k \otimes k \stackrel{mult}{\to} k

on the underlying additive group of kk.

Examples

Counterexamples:

Applications

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

Properties

Number fields kk 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\mathbb{A}_k, which are built from the local completions of kk.

Revised on April 9, 2014 21:19:51 by Urs Schreiber (82.169.114.243)