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totally real and imaginary fields (Definition)

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$:

$\displaystyle \omega\in \mathbb{R}\Leftrightarrow \omega=\bar{\omega}$
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.



"totally real and imaginary fields" is owned by alozano.
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See Also: real and complex embeddings, examples of totally real fields, examples of ramification of archimedean places

Other names:  complex multiplication field
Also defines:  totally real field, totally imaginary field, CM-field, maximal real subfield
Keywords:  totally, real, imaginary, complex multiplication

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examples of totally real fields (Example) by alozano
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Cross-references: examples of totally real fields, root, contained, strictly, complex conjugation, pointwise, fixed, complex conjugate, negative, conjugates, number, square root, field, real number, imaginary quadratic extension, complex embeddings, real embeddings, embeddings, complex numbers, subfield, real and complex embeddings
There are 14 references to this entry.

This is version 5 of totally real and imaginary fields, born on 2003-08-29, modified 2005-03-09.
Object id is 4673, canonical name is TotallyRealAndImaginaryFields.
Accessed 7603 times total.

Classification:
AMS MSC: 12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)
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