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Hometotally real and imaginary fields

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# totally real and imaginary fields

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

Let $K$ be a subfield of the complex numbers, $\mathbb{C}$, and let $\Sigma_{K}$ be the set of all embeddings of $K$ in $\mathbb{C}$.

###### Definition 1.

With $K$ as above:

1. $K$ is a

*totally real field*if all embeddings $\psi\in\Sigma_{K}$ are real embeddings.2. $K$ is a

*totally imaginary field*if all embeddings $\psi\in\Sigma_{K}$ are (non-real) complex embeddings.3. $K$ is a

*CM-field*or*complex multiplication field*if $K$ is a totally imaginary quadratic extension of a totally real field.

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 $\omega$:

$\omega\in\mathbb{R}\Leftrightarrow\omega=\bar{\omega}$ |

Thus, a field $K$ which is fixed *pointwise* by complex
conjugation is real (i.e. strictly contained in $\mathbb{R}$). However, $K$ might not be totally real. For example, let $\alpha$ be the unique real third root of $2$. Then $\mathbb{Q}(\alpha)$ 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.

## Mathematics Subject Classification

12D99*no label found*

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