# examples of totally real fields

Examples:

1. 1.

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

 $\Sigma_{K}=\{\operatorname{Id}_{K},\sigma\}$

where $\operatorname{Id}_{K}\colon K\hookrightarrow\mathbb{C}$ is the identity map ($\operatorname{Id}_{K}(k)=k$, for all $k\in K$), whereas

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

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

2. 2.

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

 $\Sigma_{K}=\{\operatorname{Id}_{K},\sigma\}$

where $\operatorname{Id}_{K}\colon K\hookrightarrow\mathbb{C}$ is the identity map ($\operatorname{Id}_{K}(k)=k$, for all $k\in K$), whereas

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

Since $\sqrt{d}\in\mathbb{C}$ and it is not in $\mathbb{R}$, it follows that $K$ is a totally imaginary field.

3. 3.

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

 $\{\zeta_{n}^{a}\mid a\in(\mathbb{Z}/n\mathbb{Z})^{\times}\}$

but those are all imaginary. Thus $\psi(L)\nsubseteq\mathbb{R}$. Hence $L$ is a totally imaginary field.

4. 4.

In fact, $L$ as in $(3)$ is a CM-field. 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

 $\zeta_{n}^{2}+\zeta_{n}^{-2}-2=2\cos(\frac{4\pi}{n})-2<0$

and any other conjugate is

 $\zeta_{n}^{2a}+\zeta_{n}^{-2a}-2=2\cos(\frac{4a\pi}{n})-2<0,a\in(\mathbb{Z}/n% \mathbb{Z})^{\times}$

Hence, $L$ is a CM-field.

5. 5.

Notice that any quadratic imaginary number field is obviously a CM-field.

Title examples of totally real fields ExamplesOfTotallyRealFields 2013-03-22 13:55:05 2013-03-22 13:55:05 alozano (2414) alozano (2414) 6 alozano (2414) Example msc 12D99 TotallyRealAndImaginaryFields NumberField examples of totally imaginary fields examples of CM-fields
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