# 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. 1.

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

2. 2.

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

3. 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.

 Title totally real and imaginary fields Canonical name TotallyRealAndImaginaryFields Date of creation 2013-03-22 13:55:02 Last modified on 2013-03-22 13:55:02 Owner alozano (2414) Last modified by alozano (2414) Numerical id 8 Author alozano (2414) Entry type Definition Classification msc 12D99 Synonym complex multiplication field Related topic RealAndComplexEmbeddings Related topic TotallyImaginaryExamplesOfTotallyReal Related topic ExamplesOfRamificationOfArchimedeanPlaces Defines totally real field Defines totally imaginary field Defines CM-field Defines maximal real subfield