Euclidean fieldEuclideanField2004-05-21 19:59:412008-03-12 03:29:06DefinitionEuclidean% this is the default PlanetMath preamble. as your knowledge
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\PMlinkescapeword{close}
\PMlinkescapeword{constructible}
\PMlinkescapeword{Euclidean}
\PMlinkescapeword{length}
\PMlinkescapeword{level}
\PMlinkescapeword{measure}
\PMlinkescapeword{open}
An ordered field $F$ is \emph{Euclidean} if every non-negative element $a$ ($a\geq0$) is a square in $F$ (there exists $b\in F$ such that $b^2=a$).
\section{Examples}
\begin{itemize}
\item
$\mathbb{R}$ is Euclidean.
\item$\mathbb{Q}$ is not Euclidean because $2$ is not a square in $\mathbb{Q}$ (\PMlinkname{i.e.}{Ie}, $\pm\sqrt{2}\notin \mathbb{Q}$).
\item $\mathbb{C}$ is not a Euclidean field because \PMlinkname{$\mathbb{C}$ is not an ordered field}{MathbbCIsNotAnOrderedField}.
\item
The \PMlinkname{field of real constructible numbers}{ConstructibleNumbers} is Euclidean.
\end{itemize}
A Euclidean field is an ordered Pythagorean field.
There are ordered fields that are Pythagorean but not Euclidean.