Euclidean field

An ordered field $F$ is Euclidean if every non-negative element $a$ ( $a\geq 0$ ) is a square in $F$ (there exists $b\in F$ such that $b^{2}=a$ ).

1 Examples

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$\mathbb{R}$ is Euclidean.
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$\mathbb{Q}$ is not Euclidean because $2$ is not a square in $\mathbb{Q}$ (i.e. (http://planetmath.org/Ie), $\pm\sqrt{2}\notin\mathbb{Q}$ ).
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$\mathbb{C}$ is not a Euclidean field because $\mathbb{C}$ is not an ordered field (http://planetmath.org/MathbbCIsNotAnOrderedField).
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The field of real constructible numbers (http://planetmath.org/ConstructibleNumbers) is Euclidean.

A Euclidean field is an ordered Pythagorean field.

There are ordered fields that are Pythagorean but not Euclidean.