examples of totally real fields
Here we present examples of totally real fields, totally imaginary fields and CM-fields.
Examples:
-
1.
Let K=ℚ(√d) with d a square-free positive integer. Then
ΣK={IdK,σ} where IdK:K↪ℂ is the identity map (IdK(k)=k, for all k∈K), whereas
σ:K↪ℂ,σ(a+b√d)=a-b√d Since √d∈ℝ it follows that K is a totally real field.
-
2.
Similarly, let K=ℚ(√d) with d a square-free negative integer. Then
ΣK={IdK,σ} where IdK:K↪ℂ is the identity map (IdK(k)=k, for all k∈K), whereas
σ:K↪ℂ,σ(a+b√d)=a-b√d Since √d∈ℂ and it is not in ℝ, it follows that K is a totally imaginary field.
-
3.
Let ζn,n≥3, be a primitive nth root of unity
and let L=ℚ(ζn), a cyclotomic extension. Note that the only roots of unity that are real are ±1. If ψ:L↪ℂ is an embedding, then ψ(ζn) must be a conjugate
of ζn, i.e. one of
{ζan∣a∈(ℤ/nℤ)×} but those are all imaginary. Thus ψ(L)⊈. Hence is a totally imaginary field.
-
4.
In fact, as in is a CM-field. Indeed, the maximal real subfield
of is
Notice that the minimal polynomial
of over is
so we obtain from by adjoining the square root of the discriminant
of this polynomial
which is
and any other conjugate is
Hence, is a CM-field.
-
5.
Notice that any quadratic imaginary number field is obviously a CM-field.
Title | examples of totally real fields |
---|---|
Canonical name | ExamplesOfTotallyRealFields |
Date of creation | 2013-03-22 13:55:05 |
Last modified on | 2013-03-22 13:55:05 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 6 |
Author | alozano (2414) |
Entry type | Example |
Classification | msc 12D99 |
Related topic | TotallyRealAndImaginaryFields |
Related topic | NumberField |
Defines | examples of totally imaginary fields |
Defines | examples of CM-fields |