Euclidean field

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In mathematics, a Euclidean field is an ordered field K for which every non-negative element is a square: that is, x ≥ 0 in K implies that x = y² for some y in K.

Contents 1 Properties 2 Examples 3 External links 4 References

[edit] Properties

• Every Euclidean field is an ordered Pythagorean field, but the converse is not true.

[edit] Examples

• The rational numbers Q with the usual operations and ordering do not form a Euclidean field. For example, 2 is not a square in Q since the square root of 2 is irrational.
• The real numbers R with the usual operations and ordering form a Euclidean field.
• The complex numbers C do not form a Euclidean field since they cannot be given the structure of an ordered field.

[edit] External links

• Euclidean Field at PlanetMath.

[edit] References

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