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, $ℂ$, and let ${\mathrm{\Sigma }}_K$ be the set of all embeddings of $K$ in $ℂ$.

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 (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 $\overline{\omega }$, the complex conjugate of $\omega$, equals $\omega$:

$$\omega \in ℝ\iff \omega =\overline{\omega }$$

Thus, a field $K$ which is fixed pointwise by complex conjugation is real (i.e. strictly contained in $ℝ$). 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