number field
Definition 1.
A field which is a finite extension of Q, the rational numbers, is called a number field
(sometimes called algebraic number field). If the degree of the extension
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 ℚ is a number field.
Example 2.
Let K=ℚ(√d), where d≠1 is a square-free non-zero integer and √d stands for any of the roots of x2-d=0 (note that if √d∈K then -√d∈K as well). Then K is a number field and [K:ℚ]=2. We can explictly describe all elements of K as follows:
K={t+s√d:t,s∈ℚ}. |
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=ℚ(ζn) be a cyclotomic extension of ℚ, where ζn is a primitive nth root of unity. Then K is a number field and
[K:ℚ]=φ(n) |
where φ(n) is the Euler phi function. In particular, φ(3)=2, therefore K3 is a quadratic number field (in fact K3=ℚ(√-3)). We can explicitly describe all elements of K as follows:
Kn={q0+q1ζn+q2ζ2n+…+qn-1ζn-1n:qi∈ℚ}. |
In fact, one can do better. Every element of Kn can be uniquely expressed as a rational combination of the φ(n) elements {ζan:gcd(a,n)=1, 1≤a<n}.
Example 4.
Let K be a number field. Then any subfield L with ℚ⊆L⊆K is also a number field. For example, let p be a prime number
and let F=ℚ(ζp) be a cyclotomic extension of ℚ, 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:
F+=ℚ(ζp+ζ-1p). |
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 | Regulator![]() |
Related topic | DiscriminantIdeal |
Related topic | ClassNumber2 |
Related topic | ExistenceOfHilbertClassField |
Related topic | Multiplicat |
Defines | quadratic number field |
Defines | quadratic field |