| Version 29 |
Version 28 |
| \PMlinkescapeword{close} |
\PMlinkescapeword{close} |
| \PMlinkescapeword{constructible} |
\PMlinkescapeword{constructible} |
| \PMlinkescapeword{Euclidean} |
\PMlinkescapeword{Euclidean} |
| \PMlinkescapeword{length} |
\PMlinkescapeword{length} |
| \PMlinkescapeword{level} |
\PMlinkescapeword{level} |
| \PMlinkescapeword{measure} |
\PMlinkescapeword{measure} |
| \PMlinkescapeword{open} |
\PMlinkescapeword{open} |
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| 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 $b^2=a$). |
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 $b^2=a$). |
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| \section{Examples} |
For example, $\mathbb{R}$ is Euclidean. On the other hand, $\mathbb{Q}$ is not Euclidean because 2 is not a square in $\mathbb{Q}$ (\PMlinkname{i.e.}{Ie}, $\pm\sqrt{2}\notin \mathbb{Q}$). Also, $\mathbb{C}$ is not a Euclidean field because \PMlinkname{$\mathbb{C}$ is not an ordered field}{MathbbCIsNotAnOrderedField}. |
| \begin{itemize} |
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| $\mathbb{R}$ is Euclidean. |
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| \item$\mathbb{Q}$ is not Euclidean because 2 is not a square in $\mathbb{Q}$ (\PMlinkname{i.e.}{Ie}, $\pm\sqrt{2}\notin \mathbb{Q}$). |
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| \item $\mathbb{C}$ is not a Euclidean field because \PMlinkname{$\mathbb{C}$ is not an ordered field}{MathbbCIsNotAnOrderedField}. |
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| \item |
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| The \PMlinkname{field of real constructible numbers}{ConstructibleNumbers} is Euclidean. |
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| \end{itemize} |
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As another example, the smallest subfield $\mathbb{E}$ of $\mathbb{R}$ over $\mathbb{Q}$ such that $\mathbb{E}$ is Euclidean is called the \emph{field of real constructible numbers}. |
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| A Euclidean field is an ordered Pythagorean field. |
A Euclidean field is an ordered Pythagorean field. |
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| There are ordered fields that are Pythagorean but not Euclidean. |
There are ordered fields that are Pythagorean but not Euclidean. |
