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^{}, $\u2102$, and let ${\mathrm{\Sigma}}_{K}$ be the set of all embeddings of $K$ in $\u2102$.
Definition 1.
With $K$ as above:

1.
$K$ is a totally real field if all embeddings $\psi \in {\mathrm{\Sigma}}_{K}$ are real embeddings.

2.
$K$ is a totally imaginary field if all embeddings $\psi \in {\mathrm{\Sigma}}_{K}$ are (nonreal) complex embeddings.

3.
$K$ is a CMfield 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 CMfield $K$ 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 $\overline{\omega}$, the complex conjugate^{} of $\omega $, equals $\omega $:
$$\omega \in \mathbb{R}\iff \omega =\overline{\omega}$$ 
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  20130322 13:55:02 
Last modified on  20130322 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  CMfield 
Defines  maximal real subfield 