PlanetMath: examples of totally real fields

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examples of totally real fields

(Example)

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

Examples:

Let $K=\mathbb{Q}(\sqrt{d})$ with a square-free positive integer. Then
$\displaystyle \Sigma_K=\{ \operatorname{Id}_K, \sigma\}$
where $\operatorname{Id}_K\colon K \hookrightarrow \mathbb{C}$ is the identity map ( $\operatorname{Id}_K(k)=k$ , for all $k\in K$ ), whereas
$\displaystyle \sigma\colon K \hookrightarrow \mathbb{C},\quad \sigma(a+b\sqrt{d})=a-b\sqrt{d}$
Since $\sqrt{d}\in \mathbb{R}$ it follows that is a totally real field.
Similarly, let $K=\mathbb{Q}(\sqrt{d})$ with a square-free negative integer. Then
$\displaystyle \Sigma_K=\{ \operatorname{Id}_K, \sigma\}$
where $\operatorname{Id}_K\colon K \hookrightarrow \mathbb{C}$ is the identity map ( $\operatorname{Id}_K(k)=k$ , for all $k\in K$ ), whereas
$\displaystyle \sigma\colon K \hookrightarrow \mathbb{C},\quad \sigma(a+b\sqrt{d})=a-b\sqrt{d}$
Since $\sqrt{d}\in \mathbb{C}$ and it is not in $\mathbb{R}$ , it follows that is a totally imaginary field.
Let $\zeta_n, n\geq 3$ , 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 $\psi\colon L \hookrightarrow \mathbb{C}$ is an embedding, then $\psi(\zeta_n)$ must be a conjugate of $\zeta_n$ , i.e. one of
$\displaystyle \{ \zeta_n^a \mid a \in (\mathbb{Z}/n\mathbb{Z})^{\times}\}$
but those are all imaginary. Thus $\psi(L)\nsubseteq \mathbb{R}$ . Hence is a totally imaginary field.
In fact, as in is a CM-field. Indeed, the maximal real subfield of is
$\displaystyle F=\mathbb{Q}(\zeta_n + \zeta_n^{-1})$
Notice that the minimal polynomial of $\zeta_n$ over is
$\displaystyle X^2-(\zeta_n+\zeta_n^{-1})X+1$
so we obtain from by adjoining the square root of the discriminant of this polynomial which is
$\displaystyle \zeta_n^2+\zeta_n^{-2}-2= 2\cos(\frac{4\pi}{n})-2 < 0$
and any other conjugate is
$\displaystyle \zeta_n^{2a}+\zeta_n^{-2a}-2=2\cos(\frac{4a\pi}{n})-2 < 0, a\in (\mathbb{Z}/n\mathbb{Z})^{\times}$
Hence, is a CM-field.
Notice that any quadratic imaginary number field is obviously a CM-field.