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\PMlinkescapetext{This page is under construction.} |
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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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| 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}. |
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}. |
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| A Euclidean field is an ordered Pythagorean field. |
A Euclidean field is an ordered Pythagorean field. |
| There are ordered fields that are Pythagorean but not Euclidean. |
There are ordered fields that are Pythagorean but not Euclidean. |
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