examples of totally real fields

Here we present examples of totally real fields, totally imaginaryfields and CM-fields.

Examples:

1. Let $K=\mathbb{Q}(\sqrt{d})$ with $d$ a square-free positive integer. Then

   $${\mathrm{\Sigma }}_K=\{{\mathrm{Id}}_K,\sigma \}$$

   where ${\mathrm{Id}}_K:K\hookrightarrow ℂ$ isthe identity map (${\mathrm{Id}}_K(k)=k$, for all $k\in K$),whereas

   $$\sigma :K\hookrightarrow ℂ,\sigma (a+b\sqrt{d})=a-b\sqrt{d}$$

   Since $\sqrt{d}\in ℝ$ itfollows that $K$ is a totally real field.

2. Similarly, let $K=\mathbb{Q}(\sqrt{d})$ with $d$ a square-freenegative integer. Then

   $${\mathrm{\Sigma }}_K=\{{\mathrm{Id}}_K,\sigma \}$$

   where ${\mathrm{Id}}_K:K\hookrightarrow ℂ$ isthe identity map (${\mathrm{Id}}_K(k)=k$, for all $k\in K$),whereas

   $$\sigma :K\hookrightarrow ℂ,\sigma (a+b\sqrt{d})=a-b\sqrt{d}$$

   Since $\sqrt{d}\in ℂ$ andit is not in $ℝ$, it follows that $K$ is a totally imaginaryfield.

3. Let ${\zeta }_n,n\ge 3$, be a primitive $n^{th}$ root ofunity and let $L=\mathbb{Q}({\zeta }_n)$, a cyclotomic extension. Notethat the only roots of unity that are real are $\pm 1$. If$\psi :L\hookrightarrow ℂ$ 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 allimaginary. Thus $\psi (L)⊈ℝ$. Hence $L$ is a totallyimaginary field.

4. In fact, $L$ as in $(3)$ is a CM-field. Indeed, the maximalreal subfield of $L$ is

   $$F=\mathbb{Q}({\zeta }_n+{\zeta }_n^{-1})$$

   Noticethat 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 thediscriminant of this polynomial which is

   and any other conjugate is

   Hence, $L$ is a CM-field.

5. Notice that any quadratic imaginary number field isobviously a CM-field.