number field

Definition 1.

A field which is a finite extensionMathworldPlanetmath of Q, the rational numbers, is called a number fieldMathworldPlanetmath (sometimes called algebraic number field). If the degree of the extensionPlanetmathPlanetmathPlanetmathPlanetmath K/Q is n then we say that K is a number field of degree n (over Q).

Example 1.

The field of rational numbers is a number field.

Example 2.

Let K=(d), where d1 is a square-free non-zero integer and d stands for any of the roots of x2-d=0 (note that if dK then -dK as well). Then K is a number field and [K:]=2. We can explictly describe all elements of K as follows:

Definition 2.

A number field K such that the degree of the extension K/Q is 2 is called a quadratic number field.

In fact, if K is a quadratic number field, then it is easy to show that K is one of the fields described in Example 2.

Example 3.

Let Kn=(ζn) be a cyclotomic extension of , where ζn is a primitive nth root of unityMathworldPlanetmath. Then K is a number field and


where φ(n) is the Euler phi function. In particular, φ(3)=2, therefore K3 is a quadratic number field (in fact K3=(-3)). We can explicitly describe all elements of K as follows:


In fact, one can do better. Every element of Kn can be uniquely expressed as a rational combinationMathworldPlanetmathPlanetmath of the φ(n) elements {ζna:gcd(a,n)=1, 1a<n}.

Example 4.

Let K be a number field. Then any subfieldMathworldPlanetmath L with LK is also a number field. For example, let p be a prime numberMathworldPlanetmath and let F=(ζp) be a cyclotomic extension of , where ζp is a primitive pth root of unity. Let F+ be the maximal real subfield of F. F+ is a number field and it can be shown that:

Title number field
Canonical name NumberField
Date of creation 2013-03-22 12:04:09
Last modified on 2013-03-22 12:04:09
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 17
Author alozano (2414)
Entry type Definition
Classification msc 11-00
Synonym algebraic number field
Related topic AlgebraicNumberTheory
Related topic ExamplesOfPrimeIdealDecompositionInNumberFields
Related topic ExamplesOfFields
Related topic AbelianExtensionsOfQuadraticImaginaryNumberFields
Related topic NumberTheory
Related topic ResidueDegree
Related topic RegulatorMathworldPlanetmath
Related topic DiscriminantIdeal
Related topic ClassNumber2
Related topic ExistenceOfHilbertClassField
Related topic Multiplicat
Defines quadratic number field
Defines quadratic field