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| Title of object: |
formally real field |
| Canonical Name: |
FormallyRealField |
| Type: |
Definition |
| Created on: |
2004-05-18 16:07:58 |
| Modified on: |
2004-05-18 18:56:33 |
| Classification: |
msc:12D15 |
Preamble:
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Content:
\PMlinkescapeword{real}
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
$S_F\not= F$
\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
$\mathbb{R}, \mathbb{Q}$ are all 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 over $F[x]$. As an example, $\mathbb{Q}(\omega)$ is formally real, where $\omega\not= 1$ is a fourth-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 even orderable.
\end{itemize} |
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