examples of totally real fields
Here we present examples of totally real fields, totally imaginary fields and CMfields.
Examples:

1.
Let $K=\mathbb{Q}(\sqrt{d})$ with $d$ a squarefree positive integer. Then
$${\mathrm{\Sigma}}_{K}=\{{\mathrm{Id}}_{K},\sigma \}$$ where ${\mathrm{Id}}_{K}:K\hookrightarrow \u2102$ is the identity map (${\mathrm{Id}}_{K}(k)=k$, for all $k\in K$), whereas
$$\sigma :K\hookrightarrow \u2102,\sigma (a+b\sqrt{d})=ab\sqrt{d}$$ Since $\sqrt{d}\in \mathbb{R}$ it follows that $K$ is a totally real field.

2.
Similarly, let $K=\mathbb{Q}(\sqrt{d})$ with $d$ a squarefree negative integer. Then
$${\mathrm{\Sigma}}_{K}=\{{\mathrm{Id}}_{K},\sigma \}$$ where ${\mathrm{Id}}_{K}:K\hookrightarrow \u2102$ is the identity map (${\mathrm{Id}}_{K}(k)=k$, for all $k\in K$), whereas
$$\sigma :K\hookrightarrow \u2102,\sigma (a+b\sqrt{d})=ab\sqrt{d}$$ Since $\sqrt{d}\in \u2102$ and it is not in $\mathbb{R}$, it follows that $K$ is a totally imaginary field.

3.
Let ${\zeta}_{n},n\ge 3$, be a primitive ${n}^{th}$ root of unity^{} and let $L=\mathbb{Q}({\zeta}_{n})$, a cyclotomic extension. Note that the only roots of unity that are real are $\pm 1$. If $\psi :L\hookrightarrow \u2102$ is an embedding, then $\psi ({\zeta}_{n})$ must be a conjugate^{} of ${\zeta}_{n}$, i.e. one of
$$\{{\zeta}_{n}^{a}\mid a\in {(\mathbb{Z}/n\mathbb{Z})}^{\times}\}$$ but those are all imaginary. Thus $\psi (L)\u2288\mathbb{R}$. Hence $L$ is a totally imaginary field.

4.
In fact, $L$ as in $(3)$ is a CMfield. Indeed, the maximal real subfield^{} of $L$ is
$$F=\mathbb{Q}({\zeta}_{n}+{\zeta}_{n}^{1})$$ Notice that the minimal polynomial^{} of ${\zeta}_{n}$ over $F$ is
$${X}^{2}({\zeta}_{n}+{\zeta}_{n}^{1})X+1$$ so we obtain $L$ from $F$ by adjoining the square root of the discriminant^{} of this polynomial^{} which is
$$ and any other conjugate is
$$ Hence, $L$ is a CMfield.

5.
Notice that any quadratic imaginary number field is obviously a CMfield.
Title  examples of totally real fields 

Canonical name  ExamplesOfTotallyRealFields 
Date of creation  20130322 13:55:05 
Last modified on  20130322 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 CMfields 