formally real fieldFormallyRealField2004-05-18 16:07:582009-03-20 21:04:49Definitionformally real% this is the default PlanetMath preamble. as your knowledge
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A field $F$ is called \emph{formally real} if $-1$ can not be expressed as a sum of squares (of elements of $F$).
Given a field $F$, let $S_F$ be the set of all sums of squares in $F$. The following are equivalent conditions that $F$ is formally real:
\begin{enumerate}
\item
$-1\notin S_F$
\item
$S_F\not= F$ and $\operatorname{char}(F)\ne 2$
\item
$\sum {a_i}^2=0$ implies each $a_i=0$, where $a_i\in F$
\item
$F$ can be ordered (There is a total order $<$ which makes $F$ into an ordered field)
\end{enumerate}
\textbf{Some Examples:}
\begin{itemize}
\item
$\mathbb{R}$ and $\mathbb{Q}$ are both formally real fields.
\item
If $F$ is formally real, so is $F(\alpha)$, where $\alpha$ is a root of an irreducible polynomial of odd degree in $F[x]$. As an example, $\mathbb{Q}(\sqrt[3]{2}\omega)$ is formally real, where $\omega\not= 1$ is a third root of unity.
\item
$\mathbb{C}$ is not formally real since $-1=i^2$.
\item
Any field of characteristic non-zero is not formally real; it is not \PMlinkescapetext{even} orderable.
\end{itemize}