number field (original) (raw)

Definition 1.
Example 1.
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:

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 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=ℚ⁢(-3)). We can explicitly describe all elements of K as follows:

Kn={q0+q1⁢ζn+q2⁢ζn2+…+qn-1⁢ζnn-1:qi∈ℚ}.

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, 1≤a<n}.

Example 4.