totally real and imaginary fieldsTotallyRealAndImaginaryFields2003-08-29 22:26:212005-03-09 17:47:17Definitiontotally real fieldtotally imaginary fieldCM-fieldmaximal real subfieldtotallyrealimaginarycomplex multiplication% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Rats}{\mathbb{Q}}For this entry, we follow the notation of the entry real and
complex embeddings.
Let $K$ be a subfield of the complex numbers, $\Complex$, and let
$\Sigma_K$ be the set of all embeddings of $K$ in $\Complex$.
\begin{defn}
With $K$ as above:
\begin{enumerate}
\item $K$ is a \emph{totally real field} if all embeddings
$\psi\in \Sigma_K$ are real embeddings.
\item $K$ is a \emph{totally imaginary field} if all embeddings
$\psi\in\Sigma_K$ are (non-real) complex embeddings.
\item $K$ is a \emph{CM-field} or \emph{complex multiplication
field} if $K$ is a totally imaginary quadratic extension of a totally real
field.
\end{enumerate}
\end{defn}
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
$\omega$:
$$\omega\in \Reals \Leftrightarrow \omega=\bar{\omega}$$
Thus, a field $K$ which is fixed \emph{pointwise} by complex
conjugation is real (i.e. strictly contained in $\Reals$). However, $K$ might not be {\it totally real}. For example, let $\alpha$ be the unique real third root of $2$. Then $\Rats(\alpha)$ is real but not totally real. \\
Given a field $L$, the subfield of
$L$ fixed pointwise by complex conjugation is called the
\emph{maximal real subfield of} $L$.
For examples (of $(1),(2)$ and $(3)$), see examples of totally real fields.