Euclidean field (original) (raw)

An ordered field F is Euclidean if every non-negative element a (a≥0) is a square in F (there exists b∈F such that b2=a).

1 Examples

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  ℝ is Euclidean.
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  ℚ is not Euclidean because 2 is not a square in ℚ (i.e. (http://planetmath.org/Ie), ±2∉ℚ).
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  ℂ is not a Euclidean field because ℂ is not an ordered field (http://planetmath.org/MathbbCIsNotAnOrderedField).
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There are ordered fields that are Pythagorean but not Euclidean.