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), $\\pm\\sqrt{2}\otin\\mathbb{Q}$ ).
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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.

单词	EuclideanField
释义	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 • $\\mathbb{R}$ is Euclidean. • $\\mathbb{Q}$ is not Euclidean because $2$ is not a square in $\\mathbb{Q}$ (i.e. (http://planetmath.org/Ie), $\\pm\\sqrt{2}\otin\\mathbb{Q}$ ). • $\\mathbb{C}$ is not a Euclidean field because $\\mathbb{C}$ is not an ordered field (http://planetmath.org/MathbbCIsNotAnOrderedField). • 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.
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