Euclidean field
An ordered field $F$ is Euclidean if every nonnegative element $a$ ($a\ge 0$) 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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$\u2102$ is not a Euclidean field because $\u2102$ 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.
Title  Euclidean field 

Canonical name  EuclideanField 
Date of creation  20130322 14:22:39 
Last modified on  20130322 14:22:39 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  34 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 12D15 
Related topic  ConstructibleNumbers 
Related topic  EuclideanNumberField 
Defines  Euclidean 