For this entry, we follow the notation of the entry real and complex embeddings.

Let $K$ be a subfield of the complex numbers, $\Complex$ and let $\Sigma_K$ be the set of all embeddings of $K$ in $\Complex$

Note that, for example, one can obtain a CM-field $K$ from a totally real number field $F$ by adjoining the square root of a number all of whose conjugates are negative.

Note: A complex number $\omega$ is real if and only if $\bar{\omega}$ the complex conjugate of $\omega$ equals $\omega$ $$\omega\in \Reals \Leftrightarrow \omega=\bar{\omega}$$ Thus, a field $K$ which is fixed pointwise by complex conjugation is real (i.e. strictly contained in $\Reals$ . However, $K$ might not be totally real. For example, let $\alpha$ be the unique real third root of $2$ Then $\Rats(\alpha)$ is real but not totally real.

Given a field $L$ the subfield of $L$ fixed pointwise by complex conjugation is called the maximal real subfield of $L$

For examples (of $(1),(2)$ and $(3)$ , see examples of totally real fields.